Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions
Zheng Liu Giovanni Rosso








































































































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Non-cuspidal Hida theory and trivial zeros]Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions


mailto:zliu@math.ucsb.eduzliu@math.ucsb.edu
<https://math.ucsb.edu/ zliu>
University of California, Santa Barbara, CA, United States



mailto:giovanni.rosso@concordia.cagiovanni.rosso@concordia.ca
<https://sites.google.com/site/gvnros/>
Concordia University, Departments of Mathematics and Statistics,
Montreal, Québec, Canada

[2010]Primary: 11F46; Secondary: 11F33, 11R23, 11S40



We study the derivative of the standard p-adic L-function associated with a P-ordinary Siegel modular form (for P a parabolic subgroup of (n)) when it presents a semi-stable trivial zero. This implies part of Greenberg's conjecture on the order and leading coefficient of p-adic L-functions at such trivial zero. We use the method of Greenberg–Stevens. For the construction of the improved p-adic L-function we develop Hida theory for non-cuspidal Siegel modular forms.

[
    Giovanni Rosso
    
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§ INTRODUCTION


In the seminal paper <cit.> the three authors consider an elliptic curve E and a prime p such that E has split multiplicative reduction at p (for example E=X_0(11) and p=11). In  this case the p-adic L-function _p(s,E) presents a trivial zero at s=1 because of the modified Euler factor at p. If the complex L-value L(1,E) is not vanishing, they conjecture that the first derivative of the p-adic L-function at s=1 is the algebraic part of the complex L-value, up to an error factor of the form log_p(q_E)/ord_p(q_E),  which they call the ℓ-invariant. Here q_E is the Tate period of E.

This conjecture has been proved in <cit.> using Hida theory and a two-variable p-adic L-function together with a one-variable improved p-adic L-function. At the same time, Greenberg generalized the conjecture of Mazur–Tate–Taitelbaum to the class of p-adic Galois representations V that satisfy the so-called Panchishkin condition. Assuming L(0,V)≠ 0, his conjecture, roughly speaking, predicts that the multiplicity of the trivial zero of _p(s,V) at s=0 equals the order of vanishing of _p(s,V) at s=0, and gives an exact formula for the leading coefficient of the p-adic L-function. This precise formula involves a factor ℓ(V), called the ℓ-invariant of V, which is defined in purely Galois theoretic terms and coincides with log_p(q_E)/ord_p(q_E) when V is the Tate module of an elliptic curve E. This conjecture has been recently generalized to all semi-stable Galois representations (which satisfy some technical conditions) <cit.>. For the precise statement, see Conjecture <ref>. 


Let n be an integer and let P be the parabolic of GL(n) associated with the partition n=n_1+…+n_d, i.e.

    P   ={.[ a_1   *   *;   ⋱   *; a_d ]∈GL(n) | a_i∈GL(n_i), 1≤ i≤ d}.

The main objective of the paper is to study Conjecture <ref> when V is the standard Galois representation associated to an irreducible cuspidal automorphic representation π of Sp(2n,) which is P-ordinary, i.e.  the archimedean component π_∞ is isomorphic to a holomorphic discrete series _ of weight =(t^P_1,…, t^P_1_n_1,t^P_2,…, t^P_2_n_2,…,t^P_d,…, t^P_d_n_d), and the action of certain ^P_p-operators (which are Hecke operators at p whose normalization depends on ) on π admits a non-zero eigenvector with eigenvalues being p-adic units. 

Denote by L(s,π×ξ) the standard L-function for π twisted by a finite order Dirichlet character ξ. It is defined as an infinite Euler product. The local L-factor for a place v where both π and ξ are unramified is given as

    L_v(s,π_v×ξ_v)=(1-ξ_v(q_v)q_v^-s)^-1∏_i=1^n(1-ξ_v(q_v)α_v,iq_v^-s)^-1(1-ξ_v(q_v)α_v,i^-1q_v^-s)^-1,
  
where α_v,i^± 1, 1≤ i≤ n, are the Satake parameters of π_v and q_v is the cardinality of the residue field.  
The Deligne critical points for L(s,π×ξ) are the integers s_0 such that 

    1≤ s_0≤ t_d^P-n, (-1)^s_0+n=ξ(-1), or  n+1-t_d^P≤ s_0≤ 0,  (-1)^s_0+n+1=ξ(-1).

The algebracity of these critical L-values divided by certain Petersson norm period has been shown in <cit.>. In <cit.>, the first author constructed an n+1-variable p-adic L-function interpolating the critical values   to the right of the center of the partial standard L-function with π varying in a Hida family (ordinary for P=B the Borel of GL(n)).


In this paper we generalize the results of <cit.> and construct a d+1-variable p-adic L-function for P-ordinary Hida families (where P is general), interpolating critical values to the left of the center of the partial standard L-function. Moreover, we improve the formulae on the interpolation properties of the constructed p-adic L-functions by using results in <cit.> and functional equation for local doubling zeta integrals. The modified Euler factors at the place p and the archimedean place appearing in the interpolation formulae match exactly with the prediction in <cit.>. The L-functions appearing in the interpolation formula are not  the motivic L-functions but some imprimitive version, as the Euler factors at the bad primes are missing.  Still, our `imprimitive'  standard p-adic L-functions for P-ordinary Siegel modular forms of general vector weights verify the `imprimitive' version of the conjecture of Coates–Perrin-Riou.

Let T_P=P/SP be the maximal quotient torus of P. We say that ^P∈Hom_cont(T_P( Z_p), Q_p^×) is arithmetic if it is a product of an algebraic character corresponding to integers (t_1^P,…,t_d^P) and a finite order character (ϵ_1^P,…,ϵ_d^P); we say that it is admissible if moreover t_1^P ≥  t_2^P …≥ t^P_d≥ n+1. We fix a sufficiently large p-adic field F as coefficient field and denote by O_F its valuation ring. 

Suppose p≥ 3. Hida theory for P-ordinary cuspidal Siegel modular forms has been developed in <cit.> generalizing the case P=B in <cit.>. Let _P be a geometrically irreducible component of the spectrum of  ^0,N_, the Hecke algebra acting on P-ordinary Hida families of cuspidal Siegel modular forms of tame principal level N, and let F__P be its function field. We denote by __P the integral closure of Λ_P := O_F  T_P( Z_p)^∘ in F__P, where T_P(_p)^∘ is the maximal p-profinite subgroup of T_P(_p). Let (n,)^* be the subset of (n,) consisting of elements α such that Tr (α a) ∈ for all symmetric matrices a ∈(n,). Let  (n,)^*_≥ 0 be the subset of semi-positive definite matrices and (n,)^*_>0 the subset of positive definite matrices.

We prove the following theorem: 
[Theorem <ref>]
Let _P be as above. For a Dirichlet character ϕ with conductor dividing N and ϕ^2≠ 1, a pair (β_1,β_2)∈ N^-1(n,)^*⊕ 2_>0, and j∈/(p-1) such that ϕω^j(-1)=1, there is a p-adic L-function __P,ϕω^j,β_1,β_2∈__P[[S]] ⊗___P F__P with the following interpolation property. 

Let x:__P F' be an F'-point of _P (with F' being a finite extension of F). Suppose that the weight  map Λ_P ^0,N_  is étale at x  and maps x to an admissible point ^P∈_(T_P(_p),F^'×). For an integer n+1≤ k≤ t^P_d and a finite order character χ^∘:^×_p^× trivial on (/p)^×, if x is classical we have

    __P,ϕω^j,β_1,β_2(χ^∘(1+p)(1+p)^k-1,x) =C_x,β_1,β_2,ϕ,N
       × E_p(n+1-k,π_x×ϕχ^∘ω^j-k) E_∞(n+1-k,π_x×ϕχ^∘ω^j-k) L^Np∞(n+1-k,π_x×ϕχ^∘ω^j-k),

and 0 otherwise. 
Here the factor E_p(n+1-k,π_x×ϕχ^∘ω^j-k) (resp. E_∞(n+1-k,π_x×ϕχ^∘ω^j-k) is the modified Euler factor at p (resp. ∞) as predicted by Coates–Perrin-Riou <cit.>. (See (<ref>)(<ref>) for explicit formulae for these two factors in our case.) The constant C_x,β_1,β_2,ϕ,N is defined in Theorem <ref>, and is independent of the cyclotomic variable k, χ^∘

The construction of this p-adic L-function uses the doubling method <cit.>.

For the whole paper, we assume that ϕ^2 ≠ 1. This hypothesis is absolutely not necessary, but when ϕ^2=1 the p-adic L-function could have a possible pole in the cyclotomic variable (outside the range of interpolation), which comes from the pole of the Kubota–Leopoldt p-adic function appearing in the Fourier coefficients of the Siegel Eisenstein series on Sp(4n). When n=1 this pole cancels out if and only if _P has no CM <cit.>. In general we expect a cyclotomic pole if and only if the standard representation associated with _P is reducible and the trivial representation appears as a sub-quotient of it.


When n_d=1 and ϵ^P_d is trivial, the factor 1-ϕ_p(p)^-1α_n,x^-1p^s-1 appears in E_p(s,π_x×ϕ), where α_n,x is an algebraic number related with the _p^P-eigenvalues (see <ref> for the precise definition). Supposing that x is classical, then α_n,x corresponds to the Frobenius eigenvalue of p-adic valuation -(t^P_d-n) in the Weil representation associated to π_x,p. If ϕ(-1)=(-1)^n+1, ϕ_p(p)=1, and α_n,x_0=p^-1 for a classical point x_0∈_P(F), then the factor 1-ϕ_p(p)^-1α_n,x_0^-1p^k-n vanishes if k=n+1, and a trivial zero occurs at the point ((1+p)^n+1-1,x_0) for __P,ϕω^n+1,β_1,β_2. Denote by ρ_x_0:G_(2n+1,_p) the Galois representation attached to x_0 <cit.>. We shall call ((1+p)^n+1-1,x_0) a semi-stable trivial zero for __P,ϕω^n+1,β_1,β_2 if furthermore Fil^0.ρ_x_0|_G__p/Fil^2.ρ_x_0|_G__p is a two dimensional indecomposable G__p-representation (see <ref> for the definition of this filtration). When n_d=1 and π_x_0 is P-ordinary with α_n,x_0=p^-1, the condition on .ρ_x_0|_G__p is expected to be always satisfied (see Remark <ref>). It is for this special type of trivial zeros that we can use the Greenberg–Stevens method to study the derivative of the p-adic L-function.

The step of expressing the ℓ-invariant in terms of the derivative of ^P_p-eigenvalues in the Greenberg–Stevens method for the so-called trivial zero of type M (as named in <cit.>) has already been done <cit.>. The other step in the method, which  relates the derivative with respect to the cyclotomic variable of the p-adic L-function to the derivative with respect to the weight variable of the ^P_p-eigenvalues, applies in the following situation. Suppose that there is a d+1-variable p-adic L-function (S,T_1,…,T_d) with S as the cyclotomic variable, and it has a trivial zero at (s_0,t_1,…,t_d). If there exists a d-tuple integer (a_1,…,a_d)≠ 0, and integers a_0≠ a'_0, such that (S,T_1,…,T_d) vanishes along the line (s_0,t_1,…,t_d)+S(a_0,a_1,…,a_d) and can be improved (in the sense of saving the factor that causes the trivial zero in the interpolation result) along the line (s_0,t_1,…,t_d)+S(a'_0,a_1,…,a_d), then the strategy applies.

In our above mentioned case of the semi-stable zero, the assumption on .ρ_x_0|_G__p implies that the trivial zero is of type M. The p-adic L-function __P,ϕω^n+1,β_1,β_2 vanishes along the hyperplane S=(1+p)^n+1-1 (because of the missing factor 1-ϕ_p(p)p^-s for π with π_p unramified). Meanwhile,  when k=t^P_d the factor 1-ϕ_p(p)^-1α_n^-1p^n-k is a p-adic analytic function as α_n p^t_d^P-n can be expressed in terms of U^P_p eigenvalues. Hence there is the possibility to improve the p-adic L-function along the hyperplane S=(1+p)^t^P_d-1. The lines ((1+p)^n+1-1,x_0)+S(0,0,…,0,1) and ((1+p)^n+1-1,x_0)+S(1,0,…,0,1) satisfy the conditions in the previous paragraph.

Now in order to carry out the Greenberg–Stevens method, we need to construct the improved p-adic L-function. Indeed, by a different choice of the local sections at p for the Siegel Eisenstein series on Sp(4n) (compare the tables in <ref>), we obtain a new Eisenstein series such that applying to it the pullback formula from the doubling method produces the complex L-function without the factor 1-ϕ_p(p)^-1α_n^-1p^s-1.  

However, a new difficulty arises. One useful fact about the sections selected for constructing the p-adic L-function in Theorem <ref> is that the restrictions to Sp(2n,)×Sp(2n,) of the corresponding Siegel Eisenstein series are cuspidal, so Hida theory for cuspidal Siegel modular forms can be applied to finish the construction. However, the new Eisenstein series for the improved p-adic L-function does not restrict to (p-adic) cuspidal forms on Sp(2n,)×Sp(2n,). Therefore, Hida theory for non-cuspidal Siegel modular forms needs to be developed in order to construct the improved p-adic L-function. Such a theory has been developed for Siegel modular forms with P=GL(n) <cit.>, and for U(2,2) <cit.> which is later generalized to U(n,1) <cit.>.
In the first section we develop Hida theory for p-adic Siegel modular forms vanishing  along the strata of the toroidal compactification associated with cusp labels of rank strictly bigger than r, for an integer r ≤ n_d. 

Let ω_ be the coherent sheaf whose global sections are weight t holomorphic Siegel modular forms. Compared to <cit.>, we bypass introducing the subsheaf ω^♭_ inside ω_ and proving the base change property for its global sections. Instead, our approach is based on a careful analysis of the quotient  . V^SP,r_m,l/ V^SP,r-1_m,l., where V^SP,r_m,l (resp. V^SP,r-1_m,l) denotes the space of functions on the l-th layer of the Igusa tower modulo p^m which vanish along the strata of the toroidal compactification associated with cusp labels of rank strictly bigger than r (resp. r-1). We denote by ^SP,r the direct limit over l and m of V^SP,r_m,l. This allows us to define a useful subspace V^SP,r,♭_m,l⊂ V^SP,r_m,l, and after taking the limit, to establish the exact sequence 

    0^SP,r-1^SP,r,♭⊕_V∈_/Γ
    rk V=r_p[[T_P(_p)]]⊗__p[[T_P_n-r(_p)]]^SP_n-r,0_V 0,

from which one can establish Hida theory for ^SP,r by using  cuspidal Hida theory and induction on r. 

The idea of using exact sequences involving non-cupidal Siegel modular forms and Siegel modular forms of lower genus also appears in <cit.>. As they work in characteristic 0, for an irreducible algebraic representation W of GL(n), and a congruence subgroup L⊂(n) consisting of elements of the form [ I_n_r     ∗;     0     ∗ ], one has

    W(R)^L   =W()^L⊗ R,     for  a -algebra R.
  
However, (<ref>) fails if  is replaced by _p. The failure of equation (<ref>)  causes the difficulty for directly generalizing the  Hida theory for cuspidal Siegel modular forms to non-cuspidal Siegel modular forms. The sheaf ω^♭_ in <cit.> is about remedying the failure of (<ref>) when  is replaced by _p. This issue is bypassed in our approach, as we study the space ^SP,r via the terms on the two ends of the exact sequence (<ref>).  


Our results are summarized as follows:
[Theorem <ref>]
For the given parabolic subgroup P⊂(n)  and an integer 1≤ r≤ n_d, the following holds:
 

  * An ordinary projector e_P=e_P^2 can be defined on ^SP,r, and the Pontryagin dual of its ordinary part 

    ^r,*_=__p(e_P^SP,r,_p/_p)

(which is naturally an _F[[T_P(_p)]]-module) is finite free over Λ_P=_F[[T_P(_p)^∘]].

  * Define

    ^r_=_Λ_P(^r,*_,Λ_P).

Given a dominant arithmetic weight ^P∈_(T_P(_p),_p^×) with dominant algebraic part ^P∈ X(T_P)^+ and finite order part ^P∈_(T_P(_p),^×), let _^P be the corresponding prime ideal of _F[[T_P(_p)]]. Then

    .^r_⊗__F[[T_P(_p)]]_F[[T_P(_p)]]/_^P.∼_m_l e_P V^SP,r_m,l[^P],

and (see (<ref>), (<ref>) and (<ref>) for the definition of the congruence subgroup Γ_SP(p^l)⊂(2n,) and weight ι(^P)∈ X(T) associated to ^P∈ X(T_P))

    _l e_PM^r_(^P)(Γ∩Γ_SP(p^l),^P;F)(.^r_⊗__F[[T_P(_p)]]_F[[T_P(_p)]]/_^P.)[1/p].

Here the maps are equivariant under the action of the unramified Hecke algebra away from Np and the ^P_p-operators.


  * There is the following so-called fundamental exact sequence (in the study of Klingen Eisenstein congruence), 

    0^r-1_^r_⊕_V∈_/Γ
    rk V=r^0_V,P_n-r- ⊗__F[[T_P_n-r(_p)]]_F[[T_P(_p)]] 0,

and ^0_V,P_n-r- is the _F[[T_P_n-r(_p)]]-module of families of p-adic ordinary Siegel modular forms of degree n-r over Y_V, for the parabolic P_n-r⊂(n-r) defined by the partition n-r=n_1+…+n_d-1+(n_d-r).



The whole Section <ref> is devoted to the proof of this theorem, see in particular <ref>. 
The method can be generalized to other PEL type Shimura varieties, both with (using <cit.>) and without (using <cit.>) ordinary locus. 

With the new choice of sections at p for the Siegel Eisenstein series and the Hida theory for non-cuspidal Siegel modular forms  we can then construct the improved p-adic L-function:
[Theorem <ref>]
With the same notation as above, assume that the parity of _P is compatible with ϕ (i.e. ϕ(-1)=^P_d(-1) for a ^P in the image of the projection of _P to the weight space). There is a p-adic L-function ^__P,ϕ,β_1,β_2(x) ∈ F__P which satisfies the following interpolation property. If x is étale, its projection ^P in the weight space is admissible, and x is classical, then

    ^__P,ϕ,β_1,β_2(x)=    C_x,β_1,β_2,ϕ,N
       × E^P-imp_p(n+1-t^P_d,π_x×ϕϵ^P_d)  E_∞(n+1-t^P_d,π_x×ϕϵ^P_d)  L^Np∞(n+1-t^P_d,π_x×ϕϵ^P_d),

where  E^P-imp_p(n+1-t^P_d,π_x×ϕϵ^P_d) is the improved modified Euler factor at p defined as in <ref>.


The Greenberg–Stevens method <cit.> allows us to prove the following theorem on semi-stable trivial zeroes:
[Theorem <ref>]
Let x_0 be an F-point of _P where the weight projection map Λ_P^1,N_ is étale and maps x_0 to an admissible and classical point ^P_0 in the weight space. Suppose that the p-adic L-function __P,ϕω^n+1,β_1,β_2∈__P[[S]] ⊗___P F__P has a semi-stable trivial zero at ((1+p)^n+1-1,x_0) and the local-global compatibility is satisfied by the p-adic Galois representation ρ_x_0. (This in particular implies that ϵ^P_d is trivial.) Then we have 

    .__P,ϕω^n+1,β_1,β_2(S,x_0)/ S|_S=(1+p)^n+1-1
    
       =     -ℓ(ρ_x_0)· C_x_0,β_1,β_2,ϕ,N· E^P-imp_p(0,π_x_0×ϕ)  E_∞(0,π_x_0×ϕ)  L^Np∞(0,π_x_0×ϕ),

where ℓ(ρ_x_0) is the ℓ-invariant as defined by Greenberg.


This result almost implies the conjecture of Greenberg, up to the non-vanishing of the ℓ-invariant and of the imprimitive L-function. The non-vanishing of the ℓ-invariant is a very hard problem and it is known only in the case of <cit.>, thanks to a deep result in transcendental number theory stating that q_E is trascendental <cit.>. Note that for n=2 we know the non-vanishing of ℓ(ρ_x_0) whenever π_x_0=Sym^3(π_f_E), where f_E is the weight two modular form associated with an elliptic curve with semi-stable reduction at p. The imprimitive L-function could vanish because of the vanishing of some of the Euler factors at a prime ℓ dividing N. One may deal with such vanishing by selecting better sections at ℓ|N.



 §.§ Acknowledgments
 The authors thank Kai-Wen Lan,  Vincent Pilloni, Eric Urban for many useful discussions, and the referees for their careful reading of the paper. Part of this work has been done while GR was a Herchel Smith fellow at Cambridge University and supernumerary fellow at Pembroke College, and during many visits at Columbia University; he would like to warmly thank these institutions. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1638352.



  §.§.§ Notation
 
For the whole length of the paper we fix an odd prime p as well as isomorphism between _p and . Also, we fix a positive integer N≥ 3 prime to p and an integer n ≥ 1 together with a partition n=n_1+… +n_d with n_1,…,n_d≥ 1. 

We denote by V a free Z-module of rank 2n with a standard basis e_1, …, e_n,f_1,⋯,f_n equipped with a symplectic pairing given by [      0    I_n; -  I_n      0 ] with respect to the standard basis. Then e_1,⋯,e_n span a maximal isotropic subspace V_n inside V. Set V_n^*=V/V^_n. One can canonically identify V^*_n with the maximal isotropic subspace of V spanned by f_1,⋯, f_n and there is the polarization V=V_n⊕ V^*_n.

Let G=Sp(2n) be the algebraic group acting on V preserving the symplectic pairing. In matrix form it is

    {g∈GL(2n): g[    0  I_n; -I_n    0 ]g=[    0  I_n; -I_n    0 ]}.
 
Let Q_G be the standard Siegel parabolic subgroup of G preserving V_n, whose unipotent subgroup we denote by U_Q_G. We identify the Levi subgroup of Q_G with GL(n) via the map 

    Q_G   GL(n)
    [    a    b;    0 a^-1 ]   ⟼ a.

Denote by B the standard Borel subgroup of GL(n) consisting of upper triangular matrices, and by U_B, T its unipotent radical and maximal torus respectively. We fix the isomorphism of G_m^n with T which sends (a_1,…,a_n) to (a_1,…,a_n). The inverse image under (<ref>) of B constitutes the standard Borel subgroup B_G of G with unipotent radical N_G and maximal torus T_G. The tori T and T_G are identified via the map (<ref>).   


We put ourselves in the setting of <cit.>, i.e. considering Siegel modular forms ordinary with respect to a general parabolic subgroup of GL(n) containing B associated to our fixed partition n=n_1+…+n_d (the ordinarity considered in <cit.> is the ordinarity with respect to B). Set N_i=∑_j=1^i n_i, 1≤ i≤ d. Define 

    P   ={.[ a_1   *   *;   ⋱   *; a_d ]∈GL(n) | a_i∈GL(n_i), 1≤ i≤ d},
    
       SP   ={.[ a_1   *   *;   ⋱   *; a_d ]∈SL(n) | a_i∈SL(n_i), 1≤ i≤ d},

and U_P to be the unipotent radical of P. When the partition is taken as n=1+1+⋯+1, the group P, (resp. both SP and U_P) is just B (resp. U_B). Let T_P=P/SP and we fix the following isomorphism

    T_P=P/SP   ∼G^d_m
    [ a_1;   ⋱; a_d ]u   ⟼(det(a_1),…,det(a_d)).

Note here that T_P is not the maximal torus in P. Denote by X(T_P) the group of characters of T_P (which are also naturally viewed as characters of P). We identify it with Z^d by associating to t^P:=(t^P_1,…,t^P_d) the character sending (a_1,…,a_d) to ∏_i=1^ddet(a_i)^t^P_i. When working with B, we shall drop the superscript from the notation for the characters when there is unlikely confusion. The map (<ref>) restricts to a map T T_P, which induces an embedding

    :X(T_P)     X(T)
    t^P:=(t^P_1,…,t^P_d)    ⟼ (t^P_1,…, t^P_1_n_1,t^P_2,…, t^P_2_n_2,…,t^P_d,…, t^P_d_n_d).

Denote by X(T)^+ the subset of X(T) of dominant weights with respect to B and set X(T_P)^+=X(T_P)∩ X(T)^+. Then ^P∈ X(T_P) belongs to X(T_P)^+ if and only if t^P_1≥ t^P_2≥⋯≥ t^P_d. Fix a finite extension F of Q_p (assumed to be sufficiently large in the context) and denote by O_F its ring of integers. The weight space in Hida theory for P-ordinary Siegel modular forms over F  is Spec(O_F[[T_P(Z_p)]]). For an arithmetic point of the weight space ^P∈Spec(O_F[[T_P(Z_p)]])(Q_p), i.e. a character in Hom_cont(T_P(Z_p),Q^×_p) that is the product of an algebraic and a finite order character, we write its algebraic part (resp. finite order part) as ^P_alg=^P=(t^P_1,…,t^P_d) (resp. ^P_f=^P=(ϵ^P_1,…,ϵ^P_d)).

We fix the standard additive character _A=⊗_v_v:Q\AC^× with local component _v defined as _v(x)=e^-2π i{x}_v,    v≠∞
e^2π ix,   v=∞ where {x}_v is the fractional part of x. 




§ NON-CUSPIDAL HIDA THEORY


In this section we develop Hida theory for non-cuspidal Siegel modular forms, or more precisely, for the P-ordinary Siegel modular forms vanishing along the strata with cusp labels of rank >r for some 0≤ r≤ n_d. Later, the family of Siegel Eisenstein series on (4n) we shall use to construct the improved p-adic L-function, unlike the ones for the usual p-adic L-function, does not restrict to cuspidal forms on (2n)×(2n) (see the discussion at the end of <ref>). The Hida theory developed here will be applied to them. Also, we expect such a theory to be of independent interest and to find applications elsewhere, for instance in the study of Eisenstein congruences.

The main difficulty in directly generalizing  Hida theory for cuspidal forms on PEL Shimura varieties to non-cuspidal forms is that for an algebraic representation W of (n)_/ of finite rank, an algebra R and a subgroup L⊂(n,) of the form

    L={[ I_n-r     ∗;     0     ∗ ]∈(n,)},   1≤ r≤ n,

the module W(R/p^m)^L is not necessarily equal to W(R)^L⊗ R/p^m.

In <cit.> and <cit.>, a subsheaf of ω^♭_⊂ω_ is introduced to remedy this failure of base change property. The sheaf ω^♭_ is not free and differs from ω_ along the boundary of the toroidal compactification. The base change property for global sections of ω^♭_ is shown loc. cit. With this base change property, by mimicking Hida's method <cit.>, Hida theory for certain non-cuspidal forms on (2,2) and (n,1) is established loc. cit. 

Here we take a different approach. Let ^SP,r be the space of p-adic forms for the parabolic P vanishing along strata indexed by cusp labels of rank > r with p-power torsion coefficients. Instead of studying the space ^SP,r via the classical Siegel modular forms embedded in it through (<ref>) (for which a base change property for the space of certain non-cuspidal classical Siegel modular forms is required), we make a careful analysis of the Igusa tower over the boundary and define a nice subspace ^SP,r,♭ inside ^SP,r. The exact sequences in Proposition <ref> plus Proposition <ref> allow us to deduce desired properties for the space ^SP,r,♭ from those for the space of cuspidal p-adic Siegel modular forms. Meanwhile, Proposition <ref> shows that the desired properties for ^SP,r,♭ imply the existence of a nice ordinary projection on ^SP,r. Then we obtain Hida theory for non-cuspidal Siegel modular forms as summarized in Theorem <ref>.

Exact sequences for automorphic bundles and p-adic analytic deformation of automorphic bundles, similar to those in Proposition <ref>, are used in <cit.> and <cit.>, where things are simpler than our case here because everything is in characteristic zero and the issue of base change does not appear. 




 §.§ Compactifications of Siegel varieties


We start by briefly recalling some facts on the toroidal and minimal compactifications of Siegel varieties of principal level. We mainly follow the notation in <cit.>. Fix an integer N≥ 3 coprime to p. Let Y be the degree n Siegel variety of principal level N defined over Z[1/N,ζ_N]. All the objects we consider in the following are endowed with principal level N structure and we shall omit N to lighten the notation. 

Recall that V= Z^2n with standard basis e_1,⋯,e_n,f_1,⋯,f_n and the symplectic pairing given by [     0   I_n; - I_n     0 ].  For 1≤ r≤ n, let V_r be the submodule of V spanned by e_1,⋯,e_r (we sometimes call it the standard submodule of rank r in ), and we put V_0={0}. The group Sp( V)≅Sp(2n, Z) acts on V preserving the symplectic pairing. Denote by Γ the kernel of the projection Sp(2n, Z)Sp(2n, Z/N Z).

Denote by C_ V the set of cotorsion free isotropic Z-submodules of V. The group Sp(2n, Z) acts naturally on C_ V. The quotient C_ V/Γ is called the set of cusp labels of level Γ (or of principal level N). For a free Z-module X of finite rank, we write C(X) to denote the cone of positive semi-definite symmetric bilinear forms on X⊗R with rational radicals, and by C(X)^∘ its interior. A surjective morphism X X' of free Z-modules induces an inclusion C(X') C(X).  Define C_ V as the quotient of the disjoint union ∐_V∈C_ VC( V/V^) by the equivalence relations induced by the inclusions C( V/V^) C( V/V^') for V⊂ V', V,V'∈C_ V. The group Sp(2n, Z) acts on C_ V.

A GL(n, Z)-admissible smooth rational polyhedral cone decomposition Σ of C( Z^n) (<cit.>) gives rise to a rational polyhedral cone decomposition Σ_C_ V of C_ V. Corresponding to it is a toroidal compactification X^Σ of Y endowed with an action of Sp(2n, Z) <cit.>. 

The toroidal compactification X^Σ comes with a stratification indexed by Σ_C_ V/Γ, and we denote by Z_σ the stratum in X^Σ associated with σ∈Σ_C_ V. There is a canonical map Σ_C_ VC_ V sending σ to the unique V_σ∈C_ V satisfying σ⊂ C( V/V_σ^)^∘. The locally closed subscheme Z_V⊂ X^Σ is defined as the union ∐_σ∈Σ_C_ V/Γ,V_σ=V Z_σ. For 0≤ r≤ n, define ^r_X^Σ to be the sheaf of ideals associated to the closed subscheme ∐_V∈_/Γ,rk V>r Z_V.



Over X^Σ, there is the canonical semi-abelian scheme G_ /X^Σ whose restriction to Y is the universal principally polarized abelian scheme of genus n with principal level N structure. The coherent sheaf ω over X^Σ is defined as the sheaf of invariant differentials of G_ /X^Σ, which is locally free of rank n.

From the toroidal compactification, the minimal compactification is constructed as

    X^⋆=Proj(⊕_k≥ 0H^0(X^Σ,det^kω)).

The projection π:X^Σ X^⋆ is proper with connected fibers. The minimal compactification X^⋆ is stratified by C_ V/Γ. The stratum Y_V corresponding to V∈C_ V/Γ is defined as the image of Z_V. As a scheme it is isomorphic to the Siegel variety of degree (n-rk V) and principal level N. 











 §.§ The Igusa tower over the ordinary locus and p-adic forms
 
The invertible sheaf det^kω descends to an invertible sheaf on X^⋆, which we still denote by det^kω. For sufficiently large k, it is very ample over X^⋆. Choose k such that the k-th power of the Hasse invariant, which is an element in H^0(X^⋆_/F_p[ζ_N], det^p-1ω), lifts to a global section E∈H^0(X^⋆,det^t(p-1)ω) (for t ≥ 0 an integer big enough). We write its pull-back in H^0(X^Σ,det^t(p-1)ω) also as E.

Let X^⋆,ord=X^⋆[1/E] and X^Σ,ord=X^Σ[1/E], and define Y^_V, Z^_σ, Z^_V similarly. The reductions modulo powers of p of these schemes are independent of the choice of E and are called the ordinary loci. Note that X^⋆, is affine, while X^Σ, is not (except when n=1).

Fix a finite extension F of Q_p containing all N-th roots of unity. We regard all above schemes as defined over O_F. For m≥ 1, we use a subscript _m to indicate the reduction modulo p^m. Over X^Σ,ord_m, we consider the full Igusa tower 

    T^Σ_m,l=Isom_X^Σ,ord_m(μ^n_p^l,G_/X_m^Σ,ord[p^l]^∘),

for l≥ 1. It is an étale cover of X^Σ,ord_m with Galois group isomorphic to GL(n, Z/p^l Z). 
The group 

    Γ_0(p^l)={g=[ a_g b_g; c_g d_g ]∈ G( Z):c_g≡ 0 p^l}

naturally acts on T^Σ_m,l with g acting on the principal level N structure and a_g acting by the Galois action of GL(n, Z/p^l Z) on T^Σ_m,l over X^Σ,ord_m. 

Define

    ^Σ_SP,m,l=.T^Σ_m,l/SP( Z_p/p^l_p).

(see (<ref>) and (<ref>) for the definition of the algebraic subgroups P, SP of (n)). It parametrizes (in addition to the structure parametrized by X^Σ,ord_m) the level structure (E_i,ε_i)_1≤ i≤ d, p^l, where {0}=E_0⊂ E_1⊂⋯⊂ E_d=G[p^l]^∘ is a d-step increasing filtration and ε_i is an isomorphism ⋀^n_iμ_p^l^n_i≅⋀^n_i E_i/E_i-1. There is a natural T_P( Z_p)-action on ^Σ_SP,m,l. 



Write f_m,l:^Σ_SP,m,l X^Σ,ord_m for the natural projection. Define

    V^SP,r_m,l    =H^0(^Σ_SP,m,l,  f^*_m,lI^r_X^Σ,ord_m),
    ^SP,r    =_m_l V^SP,r_m,l.

The space ^SP,r is called the space of p-adic Siegel modular forms for the parabolic P vanishing along the strata indexed by cusp labels of rank >r with p-power torsion coefficients. When P=B we shall drop the P from the notation, and when r=n we shall drop r from the notation.


The T_P( Z_p)-action on ^Σ_SP,m,l equips V^SP,r_m,l and ^SP,r with an O_F T_P( Z_p)-module structure. The space ^0 is the space of cuspidal p-adic Siegel modular forms with p-power torsion coefficients, which is considered in Hida theory (for the Borel B) for cuspidal Siegel modular forms, while ^SP,0 (for general P) is the one in <cit.>. 

Besides the torsion Z_p-module ^SP,r (which is in fact p-divisible by Remark <ref>), we will also consider the Z_p-module _m_l V^SP,r_m,l, i.e. taking the inverse instead of direct limit with respect to m (which is torsion free over Z_p by Remark <ref>). It is the torsion Z_p-module ^SP,r that will be used to construct the O_F T_P( Z_p)-module of Hida families. Meanwhile, the space _m_l V^SP,r_m,l is more easily seen related to the classical Siegel modular forms.

More precisely, for ^P∈ X(T_P)^+, ^P∈(T_P(/p^l),^×), and ^P∈_(T_P(_p),^×_p) which is the product of ^P and ^P, there is a canonical Hecke-equivariant embedding <cit.>

    _l  M^r_(^P)(Γ∩Γ_SP(p^l),^P;F)(_m_l V^SP,r_m,l[^P])[1/p].

Here M^r_(Γ∩Γ_SP(p^l),^P;F) denotes the space of classical holomorphic Siegel modular forms of weight =(^P) and level Γ∩Γ_SP(p^l) with nebentypus ^P vanishing along strata with cusp labels of rank >r, and the congruence subgroup Γ_SP(p^l) is defined as

    Γ_SP(p^l)={g∈(2n,):g mod  p^l  belongs to  SP(/p^l)}.

The vanishing condition here for the classical Siegel modular forms is equivalent to requiring that at all cusps all the Fourier coefficients with indices of corank > r vanish. The space _m_l V^SP,r_m,l[^P] is the ^P-eigenspace for the action of T_P(_p) on _m_l V^SP,r_m,l. The construction of the embedding (<ref>) mainly relies on the Hodge–Tate map

    Hom_R( G[p^∞]^∘,μ_p^∞)⊗_ Z_pR∼ω_ G/R

for an ordinary  semi-abelian scheme G over a Z_p-algebra R. 



 §.§ The main theorem

Our goal is to establish the following theorem.

For given P⊂(n) as in (<ref>) and an integer 1≤ r≤ n_d, we have the following.
 

  * An ordinary projector e_P=e_P^2 can be defined on ^SP,r, and the Pontryagin dual of its ordinary part 

    ^r,*_=__p(e_P^SP,r,_p/_p)

(which is naturally an _F T_P(_p)-module) is finite free over Λ_P=_F T_P(_p)^∘]], where T_P(_p)^∘ is the maximal p-profinite subgroup of T_P(_p).

  * Define

    ^r_=_Λ_P(^r,*_,Λ_P).

Given an arithmetic weight ^P∈_(T_P(_p),_p^×) with dominant algebraic part ^P∈ X(T_P)^+ and finite order part ^P∈_(T_P(_p),^×), let _^P be the corresponding prime ideal of _F T_P(_p). Then

    .^r_⊗__F T_P(_p)_F T_P(_p)/_^P.∼_m_l e_P V^SP,r_m,l[^P],

which combining with (<ref>) gives

    ._l e_P  M^r_(^P)(Γ∩Γ_SP(p^l),^P;F)(^r_⊗__F T_P(_p)_F T_P(_p)/_^P)[1/p]..

Here the maps are equivariant under the action of the unramfied Hecke algebra away from Np and the ^P_p-operators.


  * There is the following so-called fundamental exact sequence (in the study of Klingen Eisenstein congruence), 

    0^r-1_^r_⊕_V∈_/Γ
    rk V=r^0_V,P_n-r- ⊗__F T_P_n-r(_p)_F T_P(_p) 0,

where 

    P_n-r={.[ a_1   *   *;   ⋱   *; a_d ]∈GL(n-r) | a_i∈GL(n_i), 1≤ i≤ d-1, a_d∈(n_d-r)},

and ^0_V,P_n-r- is the _F T_P_n-r(_p)-module of families of p-adic cuspidal ordinary Siegel modular forms of degree n-r over Y_V, for the parabolic P_n-r.




For two different P, P', there are no inclusion relations between ^r_ and ^r_P'-ord, thus in order to study certain Siegel modular forms by using Hida theory, one needs first to specify a parabolic P containing the standard Borel subgroup of GL(n) such that the Siegel modular forms under inspection are SP(_p)-invariant and P-ordinary. There is not such Hida theory (as to the authors' knowledge) that studies P-ordinary Siegel modular forms for all P simultaneously. On the contrary, when studying families of finite slope families, there is no need to specify a parabolic as the theory established for the Borel (e.g. <cit.>) treats all Siegel modular forms of finite slope.


The remaining part of this section is devoted to proving this theorem. The proof relies on a careful study of the quotient ^SP,r/^SP,r-1 and the boundary of the Igusa tower, which leads to the definition of the subspace V^SP,r,♭_m,l⊂ V^SP,r_m,l. This subspace is characterized by the vanishing along certain connected components of the Igusa tower over ∐_V∈_/Γ
rk V=rZ_V,, and plays an important role in our proof of the above theorem.


  


 §.§ The Mumford construction

We quickly recall the Mumford construction which will be used in the description of the fiber of the push-forward of the ideal sheaf ^r_X^Σ, as well as in the definition of q-expansions. 

Given a free Z-module X_r of rank r with basis x_1,⋯,x_r, set X^*_r to be its dual free Z-module with dual basis x^*_1,⋯,x^*_r. Let _n-r,m,l the Igusa tower, m,l≥ 1, over the degree (n-r) Siegel variety of principal level N and (A_/_n-r,m,l,ψ_N,_n-r,m,l,ϕ_p,_n-r,m,l) be the universal object over it. 

The extensions of A_/_n-r,m,l by the torus X^*_r⊗G_m are parametrized by Hom__n-r,m,l(X_r,A_/_n-r,m,l). Let B_X^*_r,m,l be an abelian scheme which is isogenous to 

    Hom__n-r,m,l(N^-1X_r,A_/_n-r,m,l)

via an isogeny of degree a power of p, related to the p-level structure of the Igusa tower. Given μ∈ N^-1X_r, there is tautologically a map c(μ):B_X^*_r,m,lA_/_n-r,m,l through evaluation at μ. Denote by S^2(X_r) the symmetric quotient of X_r⊗_Z X_r. Let PA_/_n-r,m,l×__n-r,m,lA_/_n-r,m,l be the Poincaré bundle and P^×A_/_n-r,m,l×__n-r,m,lA_/_n-r,m,l be its associated G_m-torsor. 


Pick a basis [μ_i⊗ν_i], 1≤ i≤ r(r-1)/2, of N^-1S^2(X_r) with μ_i,ν_i belonging to N^-1X_r.  Associated to each [μ_i⊗ν_i] there is a map

    c(μ_i)× c(ν_i):B_X^*_r,m,lA_/_n-r,m,l×__n-r,m,lA_/_n-r,m,l,

along which one can pull back the Poincaré bundle and its associated _m-torsor. Define

    M_X^*_r,m,l=∏_i(c(μ_i)× c(ν_i))^*(P^×)^⊗ N,
 
which is a torsor over B_X^*_r,m,l for the torus Hom_Z(N^-1S^2(X_r),G_m). For λ=∑_i=1^r(r-1)/2 a_i[μ_i⊗ν_i]∈ S^2(X_r), define the invertible sheaf (λ) over B_X^*_r,m,l as

    (λ)=⊗_i c(μ_i)× c(ν_i))^*^⊗ a_iN.

We have

    (_X^*_r,m,l_n-r,m,l)_*__X^*_r,m,l=⊕ _λ∈ N^-1S^2(X_r) H^0(_X^*_r,m,l, (λ)).


Now suppose σ⊂ C(X_r)^∘ is a cone generated by a set of elements that extends to a basis of the space of symmetric bilinear forms on X_r. Let M_X^*_r,m,lM_X^*_r,m,l,σ be the affine torus embedding over B_X^*_r,m,l corresponding to σ. Denote by σ^∨ the dual cone of σ consisting of elements in S^2(X_r)⊗ whose pairing with any element in σ is non-negative. Let σ^∨,∘ be the interior of σ^∨. Let ^σ_M_X^*_r,m,l,σ be the ideal sheaf inside _M_X^*_r,m,l,σ attached to the boundary of the affine torus embedding. Then

    (_X^*_r,m,l,σ_n-r,m,l)_*__X^*_r,m,l,σ   =⊕ _λ∈ N^-1S^2(X_r)∩ σ^∨ H^0(_X^*_r,m,l, (λ)),
    (_X^*_r,m,l,σ_n-r,m,l)_*^σ__X^*_r,m,l,σ   =⊕ _λ∈ N^-1S^2(X_r)∩ σ^∨,∘ H^0(_X^*_r,m,l, (λ)).





Let M_X^*_r,m,l,σ be the formal completion along the boundary of the torus embedding. The natural map X_rHom(X_r,S^2(X_r)) defines a period subgroup N^-1X_r⊂ X^*_r⊗G_m/Z[N^-1S^2(X_r)] with a polarization given by the duality between X_r and X_r^*. The Mumford construction gives a principally polarized semi-abelian scheme G_/M_X^*_r,m,l,σ,  together with a canonical principal level N structure ψ_N,can:(Z/NZ)^2nG_/M_X^*_r,m,l,σ[N] and a canonical trivialization ϕ_p,can:μ^n_p^l∼G_/M_X^*_r,m,l,σ[p^l]^∘, which comes from the level structure parametrized by _n-r,m,l, the extension data parametrized by B_X^*_r,m,l plus the fixed basis of X_r.

 




 §.§ The fiber of the push-forward to the minimal compactification

Let ^⋆_SP,m,l be the Stein factorization of ^Σ_SP,m,l X^⋆,,

    5pc^Σ_SP,m,l[r]^f_m,l[d]^π_    X^Σ,_m[d]^π
    ^⋆_SP,m,l[r]    X^⋆,_m.

The scheme ^Σ_SP,m,l (resp. ^⋆_SP,m,l) can also be viewed as the partial toroidal (resp. minimal) compactification of the Igusa tower _SP,m,l over Y^_m, which is a special case of the construction in <cit.>. They admit a similar description as X^Σ, X^⋆.

Let C_V, p^l⊂_ be the orbit of {V_0,V_1,⋯,V_r} under the action of the group Γ_0(p^l) and define C_V, p^l from C_V, p^l in the same way as C_V from _. The partial compactification  ^⋆_SP,m,l (resp. ^Σ_SP,m,l) is stratified by C_V, p^l/Γ∩Γ_SP(p^l) (resp. the rational polyhedral cone decomposition Σ_C_V, p^l induced from Σ). The natural maps

    _C,l:C_V,p^l/Γ∩Γ_SP(p^l)   C_V/Γ,
    _C,l:Σ_C_V,p^l/Γ∩Γ_SP(p^l)   Σ_C_V/Γ

are surjective.

In order to distinguish from the notation for the stratum indices for X^⋆, X^Σ, we use an extra  to denote the stratum indices for ^⋆_SP,m,l, ^Σ_SP,m,l, i.e. we write V, σ for elements in C_V, p^l/Γ∩Γ_SP(p^l), Σ__, p^l/Γ∩Γ_SP(p^l). The stratum in ^⋆_SP,m,l (resp. ^Σ_SP,m,l) associated to V (resp. σ, V) will be denoted as _V,m,l (resp. Z_σ,m,l, Z_V,m,l). 

The stratum _V,m,l is isomorphic to the quotient of the full level l Igusa tower over Y_V,m by Im(Γ_V∩Γ_SP(p^l)(V^/V,/p^l)), where Γ_V⊂Γ is the subgroup mapping V to itself. For V=V_σ, the diagram below describes the completion of ^Σ_SP,m,l along the stratum Z_σ,m,l,

    M_V,m,l@^(->[r][d]   M_V,m,l,σ[dl]@^(->[r]   M_V,m,l,Σ_V[dll]
    B_V,m,l[d]
    _V,m,l.

Here B_V,m,l, M_V,m,l, M_V,m,l,σ are the objects constructed in <ref> with X^*_r=V, and M_V,m,l,Σ_V is the torus embedding associated to the rational polyhedral cone decomposition Σ_V of C(/V^) given by Σ__, p^l. Let M_V,m,l,σ be the completion of M_V,m,l,Σ_V along the closure of the stratum attached to σ. Then the completion of ^Σ_SP,m,l along Z_σ,m,l is isomorphic to M_V,m,l,σ/Γ_(/V^)(p^l), where Γ_(/V^)(p^l) equals Im(Γ_V∩Γ_SP(p^l)(/V^)).

Denote by ^r_^Σ_SP,m,l (resp. ^r_^⋆_SP,m,l) the ideal sheaf attached to the union of all strata inside ^Σ_SP,m,l (resp. ^⋆_SP,m,l) with cusp labels of rank > r. The ideal sheaf ^r_^Σ_SP,m,l equals f^*_m,l^r_X^Σ,_m as f_m,l is étale. 

Since π_,*_^Σ_SP,m,l=_^⋆_SP,m,l, applying π_,∗ to the short exact sequence

    0^r_^Σ_SP,m,l_^Σ_SP,m,lι_r,*_∐_V∈_, p^l/Γ∩Γ_SP(p^l), rkV> rZ_V,m,l 0,

we get

    0π_,∗^r_^Σ_SP,m,l_^⋆_SP,m,l (π∘ι_r)_*_∐_V∈_, p^l/Γ∩Γ_SP(p^l), rkV≥ rZ_V,m,l,

where ι_r:∐_V∈_, p^l/Γ∩Γ_SP(p^l), rkV> rZ_V,m,l^Σ_SP,m,l is the canonical closed embedding. Since by definition the stratum _V,m,l is the image of Z_V,m,l, we see from (<ref>) that

    ^r_^⋆_SP,m,l=π_,∗^r_^Σ_SP,m,l.


The above description of the completion of ^Σ_SP,m,l along Z_σ,m,l, combined with (<ref>) and (<ref>), gives the following description of the fiber of the structure and ideal sheaves at a closed point x∈_V,m,l⊂^⋆_SP,m,l,

    (_^⋆_SP,m,l)^∧_x=(π_,*_^Σ_SP,m,l)^∧_x   =(⋂_σ∈ C(/V^)^∘∏_λ∈ N^-1S^2(/V^)∩ σ^∨ H^0(_V,m,l,x, (λ)))^Γ_(/V^)(p^l)
       =(∏_λ∈ N^-1S^2(/V^)_≥ 0 H^0(_V,m,l,x, (λ)))^Γ_(/V^)(p^l),

and

    (^r_^⋆_SP,m,l)^∧_x=(π_,*^r_^Σ_SP,m,l)^∧_x
    
      =   (π_,*_^Σ_SP,m,l)^∧_x∩(⋂_σ∈ C(/V^)^∘
    rk V_σ>r∏_λ∈ N^-1S^2(/V^)∩ σ^∨,∘ H^0(_V,m,l,x, (λ)))^Γ_(/V^)(p^l)
    
       =   (∏_λ∈ N^-1S^2(/V^)_≥ 0
    rk λ≥ rk V-r H^0(_V,m,l,x, (λ)))^Γ_(/V^)(p^l),

where _V,m,l,x denotes the completion of _V,m,l along its fiber over x. 


The invertible sheaf (λ) is the pull-back of an ample line bundle on a quotient of the abelian scheme _V,m,l. Thus in particular, taking the global sections commutes with base change (cf. <cit.>). Therefore (<ref>) implies that for the ideal sheaf ^r_^⋆_SP,m,l, the push-forward π_,* commutes with the base change. Since ^⋆_SP,m,l is affine, we see that the base change property holds, i.e.

    V^SP,r_m,l=V^SP,r_m+1,l⊗/p^m.






 §.§ The quotient Vr/Vr

Since X^⋆, is affine, we have

    .V^SP,r_m,l/V^SP,r-1_m,l.= H^0(X^⋆,_m, .π_* f_m,l,*^r_^Σ_SP,m,l/π_* f_m,l,*^r-1_^Σ_SP,m,l.).

We need to analyze the quotient .π_* f_m,l,*^r_^Σ_SP,m,l/π_* f_m,l,*^r-1_^Σ_SP,m,l.. It is easily seen that this quotient sheaf is supported on ∐_W∈_/Γ,rk W≥ r Y^_W,m⊂ X^⋆,_m. For x∈_W,m,l with W∈_, p^l/Γ∩Γ_SP(p^l) of rank ≥ r, using (<ref>) we get

    (.π_,*^r_^Σ_SP,m,l/π_,*^r-1_^Σ_SP,m,l.)^∧_x   =(∏_λ∈ N^-1S^2(/W^)_≥ 0
    rkλ=rkW-r H^0(_W,m,l,x, (λ)))^Γ_(/W^)(p^l)
       =(∏_V∈_, p^l/Γ∩Γ_SP(p^l)
    V⊂W, rkV=r ∏_λ∈ N^-1S^2(/W^)_≥ 0
    kerλ=V H^0(_W,m,l,x, (λ)))^Γ_(/W^)(p^l)
       =∏_V∈_, p^l/Γ∩Γ_SP(p^l)
    V⊂W, rkV=r(∏_λ∈ N^-1S^2(V^/W^)_>0 H^0(_W,V,m,l,x, (λ)))^Γ_(V^/W^)(p^l).
 
Here _W,V,m,l is the abelian scheme over _W,m,l obtained as the quotient of _W,m,l by V. It is p-power isogenous to 

    __W,m,l(N^-1(V^/W^), __W,m,l).

The invertible sheaf (λ) over _W,V,m,l with λ∈ N^-1S^2(V^/W)_>0 is defined in the way as described in <ref>. The group Γ_(V^/W^)(p^l) is the image of the stablilizer of V^/W^ inside Γ_(/W^)(p^l). 

For each V∈_, p^l/Γ∩Γ_SP(p^l), there is a closed embedding

    ι^⋆_V:^⋆_V,m,l^⋆_SP,m,l,

where ^⋆_V,m,l is the partial minimal compactification of the stratum _V,m,l. The image is the Zariski closure of the stratum _V,m,l inside ^⋆_SP,m,l, which equals the union of all strata with cusp labels containing V. Like before one can define the sheaf of ideals ^s_^⋆_V,m,l for 0≤ r≤rk V.

We define the group P^∘_n,r(/p^l) as the image of the map

    Γ_V_r∩Γ_0(p^l)   (n,/p^l)
    ccccc
       r   n-r   r   n-r
    (cccc)cα   u   ∗   ∗   r
    
       0   a   ∗   b   n-r
    
       0   0   α^-1   0   r
    
       0   c   v   d   n-r
       ⟼[ α u; 0 a ] p^l,

which is easily seen equal to [   (r,/p^l)          ∗;          0 (n-r,/p^l) ].

 
There are the following short exact sequences, 

    0π_,*^r-1_^Σ_SP,m,lπ_,*^r_^Σ_SP,m,l⊕_V∈_, p^l/Γ∩Γ_SP(p^l)
    rk V=rι^⋆_V,*^0_^⋆_V,m,l 0,


    0π_*f_m,l,*^r-1_^Σ_SP,m,lπ_*f_m,l,*^r_^Σ_SP,m,l⊕_V∈_/Γ
    rk V=r(⊕_V∈_,l^-1(V)ι^⋆_V,*^0_^⋆_V,m,l) 0,

where _,l is the projection defined in (<ref>) and

    _,l^-1(V)≃Γ_V∩Γ_0(p^l)\Γ∩Γ_0(p^l)/Γ∩Γ_SP(p^l)≃ P^∘_n,r(/p^l)\(n,/p^l)/SP(/p^l).



The short exact sequence (<ref>) follows directly from our above description in (<ref>) and (<ref>) of the fibers of the relevant sheaves on the partial minimal compactification. The term at the right end is a direct sum because the intersection between ^⋆_V,m,l and ^⋆_V',m,l, V≠V', lies inside the closed subscheme defining the ideal sheaf ^0_^⋆_Vm,l. The exact sequence (<ref>) is obtained from (<ref>) by rewriting the term at the right end. 


By taking global sections, (<ref>) gives

    0 V^SP,r-1_m,l V^SP,r_m,l⊕_V∈_/Γ
    rk V=r(⊕_V∈_,l^-1(V) H^0(^⋆_Vm,l, ^0_^⋆_Vm,l)) 0.

We see that the quotient .V^SP,r_m,l/V^SP,r-1_m,l. is a direct sum of cuspidal p-adic Siegel modular forms with p^m-torsion coefficients of level l and degree n-r with respect to certain parabolics. 

The action of the group T_P(_p) permutes the summands of .V^SP,r_m,l/V^SP,r-1_m,l.. There are too many summands in the quotient in order for it to form a nice /p^m T_P(_p)-module after taking direct limit with respect to l, or in other words the structure of the T_P(_p)-action on (<ref>) is in some sense too complicated as l grows. The idea is that we pick out a single T_P_p-orbit from   (<ref>) which patch nicely with l growing.



 §.§ The space Vr,flat

For V∈_/Γ of rank r, consider 

    _Z^_V,SP,m,l=Z^_V,m×_X^Σ_m^Σ_SP,m,l=∐_V∈_,l^-1(V)Z_V,m,l,

the restriction of the SP-Igusa tower to the stratum Z^_V,m. It is not connected if the set _,l^-1(V)≃ P^∘_n,r(/p^l)\(n,/p^l)/SP(/p^l) has more than one element. For r≤ n_d, we will define a subscheme ^♭_Z^_V,SP,m,l⊂_Z^_V,SP,m,l consisting of certain connected components which form a single orbit for the  T_P(_p)-action. The space V^SP,r,♭_m,l will be defined as the subspace of V^SP,r_m,l consisting of sections vanishing outside ^♭_Z^_V,SP,m,l.

Recall that the semi-abelian scheme G_/M_X^*_r,m,l,σ in the Mumford construction carries canonical level structures

    ψ_N,can:(Z/NZ)^2n   G_/M_X^*_r,m,l,σ[N],    ϕ_p,can:μ^n_p^l   ∼G_/M_X^*_r,m,l,σ[p^l]^∘.

We assume that if V=V_r, the standard submodule of  of rank r, and σ∈ C(/V^), then the restriction of the semi-abelian scheme (_/^Σ_SP,m,l,ψ_N,(E_i,ε_i)_1≤ i≤ d, p^l) to the formal completion along Z^_σ,m,l is isomorphic to the one induced from (G_/M_X^*_r,m,l,σ,  ψ_N,can, ϕ_p,can) (in other words, the level structures parametrized by cusps at infinity are the canonical ones).

Then for γ=[ a_γ b_γ; c_γ d_γ ]∈Γ_0(p^l), V=γ· V_r and σ=γ·σ∈ C(/V^) for some σ∈Σ, the restriction of (_/^Σ_SP,m,l,ψ_N,(E_i,ε_i)_1≤ i≤ d, p^l) to the formal completion along Z^_σ,m,l is isomorphic to the one induced from

    (G_/M_X^*_r,m,l,σ,  ψ_N,can∘γ, ϕ_p,can∘ a_γ).


If we fix V∈_,p^l, the connected components of _Z^_V,SP,m,l can be thought of in terms of the relation between the two-step filtration of _/Z^_V,m[p^∞]^∘ induced from 

    0 V⊗_m_/Z_V,m[p^∞]^∘_/Y_V×_Y_LZ_V 0,

and the d-step filtration 

    {0}=E_V,0⊂ E_V,1⊂⋯⊂ E_V,d=G_/Z^_V,m[p^l]^∘

induced from the universal object (_/^Σ_SP,m,l,ψ_N,(E_i,ε_i)_1≤ i≤ d, p^l) restricted to Z_V,m,l⊂^Σ_SP,m,l.

From now on assume r≤ n_d. Define 

    ^-1_,l(V)^♭   ={V∈^-1_,l(V):  E_V,d-1∩ V⊗μ_p^l=0},
    ^♭_Z^_V,SP,m,l   =∐_V∈^-1_,l(V)^♭ Z_V,m,l⊂_Z^_V,SP,m,l,

i.e. the union of the connected components of _Z^_V,SP,m,l for which the first d-1 steps of the parametized filtrations of _/Z^_V,m[p^l]^∘ intersect trivially with the p^l-torsion of the torus part in (<ref>).

Under the natural map

    _,l^-1(V)    SP(/p^l)\(n,/p^l)/P^∘_n,r(/p^l)
    V=γ· V_r  (γ∈Γ_0(p^l))   ⟼ a_γ,

the set ^-1_,l(V)^♭ corresponds to

    SP(/p^l)\ SP(/p^l)[     0   I_r; I_n-r     0 ]P_n,r(/p^l)/P^∘_n,r(/p^l)⊂ SP(/p^l)\(n,/p^l)/P^∘_n,r(/p^l),

with P_n,r=[      (r,/p^l) M_r,n-r(/p^l);             0    (n-r,/p^l) ]. The action of T_P(_p) on ^-1_,l(V)^♭ is transitive, and we have

    ^-1_,l(V)^♭
       ={[                                                                                                    {[     0   I_r; I_n-r     0 ]},                                                                                                                         if r< n_d,; [     0   I_r; I_n-r     0 ][               I_N_d-1                     0;                     0 (n_d,/p^l)/(n_d,/p^l) ]≃(/p^l)^×,                                                                                                                         if  r=n_d. ].















We now define ^r,♭_^Σ_SP,m,l⊂^r_^Σ_SP,m,l to be the sheaf of ideals associated to the closed subscheme given as the complement of ∐_V∈_/Γ, rkV<r_Z^_V,SP,m,l∪∐_V∈_/Γ, rkV=r^♭_Z^_V,SP,m,l inside ^Σ_SP,m,l, and define

    V^SP,r,♭_m,l    =H^0(^Σ_SP,m,l, ^r,♭_^Σ_SP,m,l)⊂ V^SP,r_m,l,
    ^SP,r,♭    =_m_l V^SP,r,♭_m,l⊂^SP,r.

If follows from the definition and (<ref>) that

    .V^SP,r,♭_m,l/V^SP,r-1_m,l.=⊕_V∈_/Γ
    rk V=r(⊕_V∈_,l^-1(V)^♭ H^0(^⋆_Vm,l, ^0_^⋆_Vm,l)).

The natural T_P(_p)-action on the left hand side induces a T_P(_p)-action on

    ⊕_V∈_,l^-1(V)^♭ H^0(^⋆_Vm,l, ^0_^⋆_Vm,l).


Let 

    P_n-r   ={.[ a_1   *   *;   ⋱   *; a_d ]∈GL(n-r) | a_i∈GL(n_i), 1≤ i≤ d-1, a_d∈(n_d-r)},
     SP_n-r   ={.[ a_1   *   *;   ⋱   *; a_d ]∈SL(n-r) | a_i∈SL(n_i), 1≤ i≤ d-1, a_d∈(n_d-r)},
     T_P_n-r   =P_n-r/SP_n-r={[       ^d_m if r< n_d,;     ^d-1_m  if r=n_d. ].

We know that for each V∈^-1_,l(V)^♭, we have

    Im(Γ_V∩Γ_SP(p^l)(V^/V,/p^l))≃Γ(N)∩ SP_n-r().


The embedding P_n-r P induces a morphism T_P_n-r T_P, and the induced action of T_P_n-r(_p) on  (<ref>) preserves each direct summand, so equips each H^0(^⋆_Vm,l, ^0_^⋆_Vm,l), V∈^-1_,l(V)^♭, with an _F T_P_n-r(_P)-module structure.

From (<ref>), we also know that for each V∈^-1_,l(V)^♭, the scheme ^⋆_V,m,l is isomorphic to the minimal compactification of the quotient by SP_n-r(/p^l) of the full level p^l Igusa tower over Y^_V,m. Denote by V^SP_n-r,0_V,m,l the space of cuspidal sections over that Igusa tower over Y^_V,m, which carries a natural T_P_n-r(_p)-action. 

Then 

    H^0(^⋆_Vm,l, ^0_^⋆_Vm,l)≃ V^SP_n-r,0_V,m,l
 
as _F T_P_n-r(_p)-modules. Furthermore, we have

    ⊕_V∈_,l^-1(V)^♭ H^0(^⋆_Vm,l, ^0_^⋆_Vm,l)≃_p T_P(_p)⊗__p T_P_n-r(_p) V^SP_n-r,0_V,m,l,

because by (<ref>) and (<ref>) the action of T_P(/p^l)/T_P_n-r(/p^l) on _,l^-1(V)^♭ is simply transitive. 

Summarizing the above discussion, if we let 
    ^SP_n-r,0_V = _m_l V^SP_n-r,0_V,m,l
  we get

There are the following short exact sequences of _F T_P(_p)-modules,

    0 V^SP,r-1_m,l V^SP,r,♭_m,l⊕_V∈_/Γ
    rk V=r_p T_P(_p)⊗__p T_P_n-r(_p) V^SP_n-r,0_V,m,l 0,


    0^SP,r-1^SP,r,♭⊕_V∈_/Γ
    rk V=r_p T_P(_p)⊗__p T_P_n-r(_p)^SP_n-r,0_V 0.






 §.§ The q-expansions

Later our analysis of the action of the ^P_p-operators on V^SP,r,♭_m,l, ^SP,r,♭ will mostly rely on q-expansions.

Specializing the construction in <ref> to the case r=n, for γ_N∈Sp(2n,Z/NZ) and a_p∈GL(n,Z_p), the evaluation at the testing object 

    (G_/M_X^*_n,m,l,σ,  ψ_N,can∘γ_N, ϕ_p,can∘ a_p),  σ∈Σ,

defines the q-expansion map 

    ε^γ_N,a_p_,m,l:V_m,l⋂_σ∈Σ_F/p^m N^-1S^2(X_n)∩σ^∨ =_F/p^m N^-1S^2(X_n)_≥ 0 .

These ε^γ_N,a_p_,m,l's glue to the q-expansion map on ,

    ε^γ_N,a_p_: F/O_F N^-1S^2(X_n)_≥ 0 =F/O_F N^-1(n,)^*_≥ 0 .

With our fixed basis x_1,⋯,x_n of X_n, we will freely identify S^2(X_n) with Sym(n,Z)^*, the set of symmetric n× n matrices with integers as diagonal entries and half-integers as off-diagonal entries, by identifying β∈Sym(n,Z) with ∑_1≤ i,j≤ nβ_ij x_i⊗ x_j. For β∈ N^-1S^2(X_n) and f∈, write ε^γ_N,a_p_(β,f) for the β-th Fourier coefficient of f, i.e. the coefficient associated with β in ε^γ_N,a_p_(f). One can check that given a∈GL(n,Z)

    ε^m(a)γ_N,aa_p_(β,f)=ε^γ_N,a_p_( aβ a,f),

where m(a)=[    a    0;    0 a^-1 ].

As illustrated in <cit.>, since the closure of every stratum associated with a cone in the toroidal compactification is irreducible and contains a stratum corresponding to a top dimensional cone, many properties of (p-adic) Siegel modular forms can be verified by examining the q-expansion. 


The method here can be generalized to other PEL type Shimura varieties. When q-expansions are not available, one needs prove the desired properties of _p-operators in <ref> by working with Fourier–Jacobi expansions.


The following two propositions give a characterization of the space ^SP,r, ^SP,r,♭ in terms of q-expansions.


Given f∈, it belongs to ^SP,r if and only if ε^γ_N,a_p_(β,f) vanishes for all γ_N∈Sp(2n, Z), a_p∈(n,_p) and β∈ N^-1S^2(X_n)_≥ 0 of rank less or equal to n-r-1. 


Given σ∈Σ_ C_ V,p^l, pick a top dimensional cone τ∈Σ_ C_ V,p^l with σ as a face. Fix an isomorphism of Spf( O_F/p^m O_F N^-1S^2(X_n)∩τ^∨) with the completion of ^Σ_SP,m,l along the point Z_τ,m,l (here for a cone in Σ_ C_ V,p^l we use the same notation to denote a corresponding cone in C(X_n)). Then the embedding of the completion of the Zariski closure of Z_σ,m,l along the point Z_τ,m,l to the completion of ^Σ_SP,m,l along Z_τ,m,l corresponds to the quotient map from O_F/p^m N^-1S^2(X_n)∩τ^∨ onto O_F/p^m N^-1S^2(X_n)∩τ^∨∩σ^, sending all β∈ N^-1S^2(X_n)∩τ^∨ that does not belong to σ^ to 0. This description shows that the vanishing condition in the proposition implies the vanishing of f along Z_σ,m,l, and the proposition follows.


In the following, by the radical of β∈ N^-1S^2(X_n)_≥ 0, we mean the sub--module of X^*_n consisting of elements that pair trivially with β via the natural map X^*_n× S^2(X_n) X_n, and by a primitive vector in X^*_n, we mean an element not divisible by p in X^*_n.


Given f∈^SP,r, it belongs to ^SP,r,♭ if and only if ε^γ_N,a_p_(β,f) vanishes for all β of corank r such that the radical of a_pβ a_p contains a primitive vector inside Z· x^*_1 + ⋯ +  Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n. 


We use the description of the completion of the Zariski closure of Z_σ,m,l along the point Z_τ,m,l given in the proof of the previous proposition, and assume that V_σ is of rank r. Identify V_n and X^*_n (together with standard basis). Take a γ∈Γ_0(p^l) such that V_τ=γ^-1· V_n and use it  to fix an isomorphism between O_F/p^m O_F N^-1S^2(X_n)∩τ^∨ and the formal completion of the structure sheaf at the point Z_τ,m,l. Then the evaluation of f at the formal neighborhood of Z_τ,m,l corresponds to the q-expansion ε^γ,a_γ_(β,f), and the E_V_τ,d-1  corresponds to the -span of a_γ(x^*_1)/p^l,…,a_γ(x^*_N_d-1)/p^l (recall that E_V_τ,d-1 is the d-1-th step of the filtration in the level structure of the SP-Igusa tower). On the other hand, the canonical two-step filtration of the semi-abelian scheme over Z_σ,m,l corresponds to V_σ⊂V_τ. Therefore the vanishing condition in the definition of ^SP,r,♭ requires the vanishing of ε^γ,a_γ_(β,f) for all β∈ N^-1S^2(X_n)∩τ^∨∩σ^ with V_σ containing a primitive element in a_γ( Z· x^*_1 + ⋯ +  Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n). Also, for a semi-positive definite β inside σ^, the radical of β equals V_σ. Hence the vanishing condition in the proposition agrees with that for defining ^SP,r,♭.





 §.§ The Up-operators

To each matrix

    γ_p,i=[   p I_i       0       0       0;       0   I_n-i       0       0;       0       0 p^-1I_i       0;       0       0       0   I_n-i ],
           1≤ i≤ n,

corresponds a Hecke operator U^P_p,i acting on ^SP. The ordinarity condition for P requires the eigenvalues of U^P_p,N_1, U^P_p,N_2,…, U^P_p,N_d to be p-adic units (recall that N_i=∑_j=1^i n_j). In <cit.>, only these U^P_p,N_1, U^P_p,N_2,…, U^P_p,N_d are introduced as they are sufficient for defining the ordinary projection in order to establish Hida theory. However, given an automorphic representation π of (2n,) generated by a holomorphic Siegel modular form ordinary for the parabolic P, in order to retrieve the full information on π_p, one needs to consider the action of all the U^P_p,i, 1≤ i≤ n (see <ref> for details). If i≠ N_1,…,N_d, the eigenvalue of U^P_p,i on P-ordinary forms is not necessarily a p-adic unit.

When shall use the expression U^P_p-operators when we do not need to specify which Hecke operators at p we are using.

Let ^∘_SP,m,l be the restriction of _SP,m,l to Y^_m⊂ X^Σ,_m. The algebraic correspondence inside ^∘_SP,m,l×^∘_SP,m,l associated to γ_p,i is defined as follows. For N_j≤ i< N_j+1, let C_i,m,l be the moduli scheme over _F/p^m parametrizing the quintuple (A,λ,ψ_N,(E_i,ε_i)_1≤ i≤ d, p^l,L), where (A,λ,ψ_N) is an ordinary abelian scheme of genus n with principal polarization λ and principal level structure ψ_N:(/N)^2n∼A[N], defined over an _F/p^m-algebra, (E_r,ε_r)_1≤ r≤ d, p^l is the structure used to define _SP,m,l in <ref>, and L⊂ A[p^2] is a Lagrangian subgroup such that rank_/pL[p]=2n-i, L[p]∩ E_j[p]=0, L[p]+E_j+1[p]=A[p]. Denote by p_1 the projection from C_i,m,l to _SP,m,l which forgets L. There is another projection p_2 sending (A,λ,ψ_N,(E_r,ε_r)_1≤ r≤ d, p^l,L) to (A/L,λ',p∘π∘ψ_N,(E'_r,ε'_r)_1≤ r≤ d, p^l), where π:A A/L is the natural isogeny, λ' is defined by π^*λ'=p^2λ, and

    E'_r   =π(E_r),    ε'_r   =π∘ε,    1≤ r≤ j   
    
       E'_r   =π(p^-1(E_r∩ p^-l+1L)),    ε'_r   =p^-min{N_r-i, n_r}π∘ε_r,    j+1≤ r≤ d.


For N_j≤ i <N_j+1, we have the following composition

    H^0(^∘_SP,m,l,_^∘_SP,m,l)p^*_2H^0(C_i,m,l,_C_i,m,l)Trp_1p^i(n+1)H^0(^∘_SP,m,l,_^∘_SP,m,l).

The image of Trp_1 belongs to p^i(n+1)H^0(^∘_SP,m,l,_^∘_SP,m,l) because the pure inseparability degree of p_1 is p^i(n+1) <cit.>. One can also check (for example by q-expansions) that such defined U^P_p,i preserves various kinds of growth conditions along the boundary, i.e. the above map restricts to a map from V^SP_m,l to p^i(n+1)V^SP_m,l. If m>i(n+1), there is a well defined map p^-i(n+1):p^i(n+1)V^SP_m,l V^SP_m-i(n+1),l. Now given f∈ V^SP_m,l, thanks to (<ref>), we can take f∈ V^SP_m+i(n+1) such that f≡f p^i(n+1), and we define

    U^P_p,i(f)=p^-i(n+1)∘Trp_1∘ p^*_2 (f).



In this section, only U^P_p,N_1, U^P_p,N_2,…, U^P_p,N_d will be used. In order to show the desired properties of their action on ^SP,r,♭, ^SP,r, we use the following proposition and Proposition <ref>, <ref> to reduce to computations on q-expansions. 




For f∈^SP, γ_N∈(2n,) and a_p∈ T(_p)⊂(n,_p), the formula on q-expansions for the action of the U^P_p-operators on f is given by

    ε^γ_N,a_p_(β, U^P_p,N_if)=∑_x∈ M_N_i,n-N_i( Z/p Z)ε^(γ^P_p,i)^-1γ_N,a_p_([  pI_N_i       0;      Nx I_n-N_i ]β[ p I_N_i      Nx;       0 I_n-N_i ], f),
 
for β∈ N^-1S^2(X_n) and 1≤ i≤ d.

One can also write down the formula for general a_p∈(n,_p) which is a little bit more complicated. We omit it here because the case a_p being diagonal suffices for our purpose thanks to (<ref>).


All the spaces ^SP,r, 0≤ r≤ n, and ^SP,r,♭, 0≤ r≤ n_d, are stable under the U^P_p-operators. 


The statement for ^SP,r follows immediately from Proposition <ref>, <ref>. By Proposition <ref>, <ref>, in order to show the statement for ^SP,r,♭, it is enough to show that if the radical of β contains a primitive vector inside Z· x^*_1 + ⋯ +  Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n, then the same holds for [  pI_N_i       0;      Nx I_n-N_i ]β[ p I_N_i      Nx;       0 I_n-N_i ]. In fact, it is not difficult to check that if v∈ Z· x^*_1 + ⋯ +  Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n is a primitive vector, then for all x∈ M_N_i,n-N_i(), 

    ·[  pI_N_i      Nx;       0 I_n-N_i ]^-1 v_β∩ X_n⊂ Z· x^*_1 + ⋯ + Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n.



Now we want to define a ^P_p-action on the quotient of the exact sequences in Proposition <ref>, and verify that the exact sequences are ^P_p-equivariant.

For V∈_ with rank r≤ n_d, we define the ^P_p-action on ^SP_n-r,0_V as follows. For γ=diag(a_1,…,a_n,a^-1_1,…,a^-1_n)∈(2n), set γ'=diag(a_1,…,a_n-r,a^-1_1,…,a^-1_n-r)∈(2n-2r). We make U^P_p,N_i act on ^SP_n-r,0_V (the space of p-adic Siegel modular forms of degree n-r for the parabolic P_n-r) by the ^P_n-r_p-operator attached to γ^'_p,N_i.


Let us denote by U^P,[N]_p⊂^P_p the subalgebra generated by the φ(N)-powers of U^P_p,N_i, 1≤ i≤ d. Here φ(N)=N·∏_q prime factors of N(1-1/q).  Rather than showing the ^P_p-equivariance of the exact sequences in Proposition <ref>, we are only able to show the U^P,[N]_p-equivariance. However, this suffices for establishing Hida theory for ^SP,r.


The exact sequences in Proposition <ref> are U^P,[N]_p-equivariant.


We show the U^P,[N]_p-equivariance of the projection _V:V^SP,r,♭_m,l V^SP_n-r,0_V,m,L, from V^SP,r,♭_m,l to the summand of .V^SP,r,♭_m,l/V^SP,r-1_m,l. indexed by V∈_,l^-1(V)^♭, by computing the q-expansions. Pick γ∈Γ_0(p^l) such that γ^-1V_n contains V (where V_n is identified with X^*_n with standard basis), and we view V as a subspace of X^*_n via γ. Then V∈_,l^-1(V)^♭ implies that V is spanned by

    (x_1,…,x_n)[ I_N_d-1       0;       0       w ][ α_1; α_2 ],

with w∈(n_d,), w≡ I_n_d N, α_1∈ M_n-r,r() and α_2∈ M_r,r()∩(r,_p).

Take s≥maxl,φ(N). There exists 

    ccc    n-r   r
    c(cc)
       n-r   A   B   
    
       r   C   D   
    ∈(n,)

with A≡[ A_1   0;   0 A_2 ] p^s, A_1∈(N_i,), A_2∈(n-r-N_i,), C≡ 0 p^s, such that

    [ A B ][ α_1; α_2 ]=0.

Define

    _V:(n-r,)   (n,)
    β'   ⟼[ I_N_d-1       0;       0    w^-1 ][ A; B ]β'[ A B ][ I_N_d-1       0;       0    w^-1 ].

Then for γ'_N∈(2n-2r,) and a'_p∈ T_n-r(_p), there exists γ_N∈(2n,), a_p∈ T(_p) such that

    ε^γ'_N, a'_p_,V(β', _V(f))=ε^γ_N, a_p_(_V(β'), f).

To prove the proposition, it suffices to check that for β'∈(n-r,)_>0 and f∈ V^SP,r,♭_m,l, 

    ε^(γ^P'_p,i)^φ(N)γ'_N, a'_p_,V(β', (U^P_p,N_i)^φ(N)_V(f))=ε^(γ^P_p,i)^φ(N)γ_N, a_p_(_V(β'), (U^P_p,N_i)^φ(N)f).

We have

    LHS of (<ref>)
    
       =   ∑_x∈ M_N_i,n-r-N_i(/p^φ(N))ε^γ'_N, a'_p_,V([ p^φ(N)I_N_i           0;          Nx   I_n-r-N_i ]β'[ p^φ(N)I_N_i          Nx;           0   I_n-r-N_i ], _V(f))
     =   ∑_x∈ M_N_i,n-r-N_i(/p^φ(N))ε^γ_N, a_p_([ A; B ][ p^φ(N)I_N_i           0;          Nx   I_n-r-N_i ]β'[ p^φ(N)I_N_i          Nx;           0   I_n-r-N_i ][ A B ], f).

Set

    x_A   =A_1^-1xA_2∈ M_N_i,n-r-N_i(/p^φ(N)),   y(x)=N^-1A^-1_1[ -I_N_i     Nx ]B∈ M_N_i,r(/p^φ(N)).
  
The map x↦ x_A is a bijection from M_N_i,n-r-N_i(/p^φ(N)) to itself. One can check that, by the definition of x_A, y(x),



    [[ p^φ(N)I_N_i          Nx           0;           0   I_n-r-N_i           0;           0           0         I_r ]^-1[ A B; C D ][ p^φ(N)I_N_i        Nx_A       Ny(x);           0   I_n-r-N_i           0;           0           0         I_r ]]^-1[ A B; C D ]∈Γ(N)∩Γ_SP(p^s),

where x,x_A,y(x) can be taken to be any lift to . Then

    p^φ(N)I_N_i   Nx
    0   I_n-r-N_iA   B=p^φ(N)I_N_i   Nx
    0   I_n-r-N_iI_n-r   0A   B
    C   D
    
       =   I_n-r   0p^φ(N)I_N_i   Nx   0
    0   I_n-r-N_i   0
    0   0   I_rA   B
    C   D
    
       =   A   Bp^φ(N)I_N_i   Nx_A   Ny(x)
    0   I_n-r-N_i   0
    0   0   I_rp^φ(N)I_N_i   Nx_A   Ny(x)
    0   I_n-r-N_i   0
    0   0   I_r^-1A   B
    C   D^-1p^φ(N)I_N_i   Nx   0
    0   I_n-r-N_i   0
    0   0   I_rA   B
    C   D
    
        =   A   Bp^φ(N)I_N_i   Nx_A   y(x)
    0   I_n-r-N_i   0
    0   0   I_r[p^φ(N)I_N_i   Nx   0
    0   I_n-r-N_i   0
    0   0   I_r^-1A   B
    C   Dp^φ(N)I_N_i   Nx_A   Ny(x)
    0   I_n-r-N_i   0
    0   0   I_r]^-1A   B
    C   D
    ∈   A   Bp^φ(N)I_N_i   Nx_A   Ny(x)
    0   I_n-r-N_i   0
    0   0   I_r·Γ(N)∩Γ_SP(p^s).

Plugging into (<ref>), we get

    LHS of (<ref>)
    
    	 =   ∑_x∈ M_N_i,n-r-N_i(/p^φ(N))ε^γ_N, a_p_(p^φ(N)I_N_i   0   0
    Nx_A   I_n-r-N_i   0
    Ny(x)   0   I_rA
    Bβ'A   Bp^φ(N)I_N_i   Nx_A   Ny(x)
    0   I_n-r-N_i   0
    0   0   I_r, f).

Next we need to use the condition f∈ V^SP,r,♭_m,l to show that its Fourier coefficient in ε^γ_N, a_p_(f) indexed by

    [ p^φ(N)I_N_i           0           0;        Nx_A   I_n-r-N_i           0;          Ny           0         I_r ][ A; B ]β'[ A B ][ p^φ(N)I_N_i        Nx_A          Ny;           0   I_n-r-N_i           0;           0           0         I_r ]

is nonzero only if y=y(x) in M_N_i, r(/p^φ(N)). By Proposition <ref>, the coefficient indexed by (<ref>) is nonzero only if the radical of (<ref>) does not contain a primitive vector inside Z· x^*_1 + ⋯ + Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n. The radical tensored with  is spanned by the columns of 

    [ p^φ(N)I_N_i        Nx_A          Ny;           0   I_n-r-N_i           0;           0           0         I_r ]^-1[ -A^-1B;    I_r ]=[ -[ p^-φ(N)Ny;         0 ]-[ p^-φ(N)I_N_i -p^-φ(N)Nx_A;            0    I_n-r-N_i ]A^-1B;                                                                                     I_r ],

which contains no primitive vector in Z· x^*_1 + ⋯ + Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n only if 

    Ny+[ I_N_i -Nx_A ]A^-1B≡ 0 p^φ(N),

and this equation is satisfied exactly when y=y(x). Therefore, the coefficient indexed by (<ref>) is nonzero only if y=y(x) in M_N_i, r(/p^φ(N)), and from (<ref>) we get

    LHS of (<ref>)
    
    	 =   ∑_x∈ M_N_i,n-r-N_i(/p^φ(N))
    y∈ M_N_i,r(/p^φ(N))ε^γ_N, a_p_(p^φ(N)I_N_i   0   0
    Nx_A   I_n-r-N_i   0
    Ny   0   I_rA
    Bβ'A   Bp^φ(N)I_N_i   Nx_A   Ny
    0   I_n-r-N_i   0
    0   0   I_r, f)
    
    	 =   ∑_x∈ M_N_i,n-N_i(/p^φ(N))ε^γ_N, a_p_(p^φ(N)I_N_i   0
    Nx   I_n-N_i_V(β')p^φ(N)I_N_i   Nx
    0   I_n-N_i,f)
    
    	 =   RHS of (<ref>).





Let s ≥ m,l. Then (U^P_p,N_d-1)^2s V^SP,r_m,l⊂ V^SP,r,♭_m,l.


By Proposition <ref>, what we need to show is that for all γ_N∈(2n,), a_p∈ T(_p), f∈ V^SP,r_m,l and β∈ N^-1S^2(X_n)_≥ 0 whose radical is of rank r and contains a primitive vector v_β inside Z· x^*_1 + ⋯ + Z· x^*_N_d-1 + p Z· x^*_N_d-1+1 +⋯+ p Z· x^*_n,

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^2sf)=0.
 
One can easily check that for all x∈ M_N_d-1,n_d(),

    ·[ p^sI_N_d-1         Nx;          0        I_r ]^-1v_β∩ X^*_n⊂· x^*_1+⋯+· x^*_N_d-1+p^s· x^*_N_d-1+1+⋯ +p^s· x^*_n,

i.e. the radical of [ p^sI_N_d-1          0;         Nx        I_r ]β[ p^sI_N_d-1         Nx;          0        I_r ] contains a primitive vector inside · x^*_1+⋯+· x^*_N_d-1+p^s· x^*_N_d-1+1+⋯ +p^s· x^*_n. Since by Proposition <ref>,

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^2sf)=∑_x∈ M_N_d-1,n_d()ε^γ^-s_p,n-1γ_N,a_p_([ p^sI_N_d-1          0;         Nx        I_r ]β[ p^sI_N_d-1         Nx;          0        I_r ], (U^P_p,N_d-1)^sf),

we reduce to showing 

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^sf)=0,

for β whose radical contains a primitive vector v_β inside Z· x^*_1 + ⋯ + Z· x^*_N_d-1 + p^s Z· x^*_N_d-1+1 +⋯+ p^s Z· x^*_n.

Write v_β=(v_β,1,…,v_β,n). Then p^s| v_β,i, N_d-1+1≤ i≤ n, and there exists 1≤ j≤ N_d-1 such that v_β,j is not divisible by p. Put w_β=(0,…,0_j-1,v_β,n,0,…,0,-v_β,j)∈^n. Then I_n-Nη· v_βw_β belongs to (n,) for all integer η. Moreover,

    I_n-Nη· v_βw_β≡ I_n+[                   0                   ⋯                   0     Nη v_β,j· v_β,1;                   ⋱                   ⋮;                   0                   ⋯                   0     Nη v_β,j· v_β,j;                   ⋱                   ⋮;                   0                   ⋯                   0 Nη v_β,j· v_β,N_d-1;                   0                   ⋯                   0                   0 ] Np^s.
 
Let x_β=v_β,j·(v_β,1,⋯,v_β,N_d-1)∈^N_d-1. Then

    [  I_n-1 Nη x_β;      0    I_r ]^-1(I_n-η· v_βw_β)≡ I_n Np^s

and

    [  p^sI_n-1 Nx+Nη x_β;         0       I_r ]^-1(I_n-η· v_βw_β)[ p^sI_n-1       Nx;        0      I_r ]∈Im(Γ∩Γ_SP(p^s)(n,)).

By definition the vector x_β∈^N_d-1 is not divisible by p. Thus we can pick C⊂ M_N_d-1,n_d(/p^s) such that

    M_N_d-1,n_d(/p^s)=(/p^s)· x_β⊕ C.

We have

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^s f)
    
       =   ∑_x∈ M_N_d-1,n_d(/p^s)ε^(γ^P_p,d-1)^-sγ_N,a_p_([ p^sI_N_d-1          0;         Nx        I_r ]β[ p^sI_N_d-1         Nx;          0        I_r ], f)
    
       =   ∑_x∈ C ∑_η∈/p^sε^(γ^P_p,d-1)^-sγ_N,a_p_([ p^sI_N_d-1          0;   Nx+Nηx_β        I_r ]β[ p^sI_N_d-1  Nx+Nη x_β;          0        I_r ], f).

Applying (<ref>), we get

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^s f)
    
       =   ∑_x∈ C ∑_η∈/p^sε^(γ^P_p,d-1)^-sγ_N,1_([ p^s I_N_d-1           0;          Nx         I_r ](I_n-η· w_βv_β)β(I_n-η· v_βw_β)[ p^sI_N_d-1         Nx;          0        I_r ], f).

Since v_β belongs to the radical of β, we know that (I_n-η· w_βv_β)β(I_n-η· v_βw_β)=β, and

    ε^γ_N,a_p_(β, (U^P_p,N_d-1)^sf)
    
       =   ∑_x∈ C ∑_η∈/p^sε^(γ^P_p,d-1)^-sγ_N,a_p_([ p^s I_N_d-1           0;          Nx         I_r ]β[ p^sI_N_d-1         Nx;          0        I_r ], f)
    
       =   ∑_x∈ Cp^s·ε^(γ^P_p,d-1)^-sγ_N,a_p_([ p^s I_N_d-1           0;          Nx         I_r ]β[ p^s I_N_d-1          Nx;           0         I_r ], f)
    
       =    0.








 §.§ Hida families of p-adic Siegel modular forms vanishing along strata with cusp labels of rank  >r

Set U^P_p=∏_i=1^d U^P_p,N_i. We first show the existence of an ordinary projector on ^SP,r by applying induction on r and using ^SP,r,♭ plus Proposition <ref>. 

For each f∈^SP,r, the limit lim_j∞(U^P_p)^j!f exists in ^SP,r.


We remark that for any endomorphism of finitely generated _F/p^m-modules, its j!-th power stabilizes when j is large enough. 

Given f inside an _F/p^m-module with an action by U^P_p, we define the following finiteness property for f.

    -4.5em
        
        The submodule generated by (U^P_p)^nφ(N)f, n≥ 0, is finitely generated over _F/p^m.
        F

It suffices to show that (F) holds for all elements in V^SP,r_m,l. For r=0 this is known thanks to <cit.>. Now assume (F) holds for V^SP,r-1_m,l. Due to Proposition <ref>, we only need to show that (F) for all f∈ V^SP,r,♭_m,l. Take f∈ V^SP,r,♭_m,l, since (F) holds for the quotient in (<ref>), there exists a_0,a_1,…,a_j_1∈_F such that

    g=(U^P_p)^(j_1+1)φ(N)f-∑_i=0^j_1 a_i(U^P_p)^iφ(N)f∈ V^SP,r-1,♭_m,l.

Then we apply (F) for (<ref>). There exists b_0,b_1,…,b_j_2∈_F such that

    (U^P_p)^(j_2+1)φ(N)((U^P_p)^(j_1+1)φ(N) f-∑_i=0^j_1 a_i(U^P_p)^iφ(N) f)
    
      =   ∑_s=0^j_2b_s(U^P_p)^sφ(N)((U^P_p)^(j_1+1)φ(N) f-∑_i=0^j_1 a_i(U^P_p)^iφ(N)f).

Therefore (U^P_p)^(j_1+j_2+2)φ(N)f belongs to the _F/p^m-span of f,(U^P_p)^φ(N)f,…,(U^P_p)^(j_1+j_2+1)φ(N)f, and (F) holds for f.


The above proposition shows that lim_j∞(U^P_p)^j! can be well defined on ^SP,r. Define the P-ordinary projector on ^SP,r as

    e_P=lim_j∞(U^P_p)^j!.

It is an idempotent projecting the space into the subspace spanned by generalized ^P_p-eigenvectors with eigenvalues being p-adic units. Similarly a P_n-r-ordinary projector can be defined for the quotient terms in the exact sequences in Proposition <ref>.

Set

    ^r_=e_P^SP,r=e_P^SP,r,♭,      ^0_V,P_n-r- =e_P_n-r^SP_n-r,0_V,  V∈ C_ V, rk V=r≤ n_d.

Applying e_P, e_P_n-r to (<ref>), we get

    0^r-1_^r_⊕_V∈_/Γ
    rk V=r_p T_P(_p)⊗__p T_P_n-r(_p)^0_V,P_n-r-  0.


Define ^r,*_ to be the Pontryagin dual of ^r_, i.e. Hom_ Z_p(^r_, Q_p/ Z_p), and similarly define ^0,*_V,P_n-r-. Then (<ref>) gives

    0⊕_V ∈ C_ V/Γ
    rk V=r^0,*_V,P_n-r- ⊗_ Z_p T_P_n-r( Z_p) Z_p T_P( Z_p)   ^r,*_^r-1,*_ 0.


Let Λ_P= O_F T_P( Z_p)^∘ (resp. Λ_P_n-r= O_F T_P_n-r( Z_p)^∘), where T_P( Z_p)^∘ is the maximal p-profinite subgroup of T_P( Z_p) (resp. T_P_n-r(_p)).

^r,*_, 0≤ r≤ n_d, is a free Λ_P-module of finite rank.


We prove the proposition by induction. For r=0 the control theorem in <cit.> for ^0,*_ (resp. ^0,*_V,P_n-r-) says that it is a free Λ_P-module (rep. Λ_P_n-r-module) of finite rank. Suppose that ^r-1,*_ is a free Λ_P-module of finite rank. Then the terms at the two ends of (<ref>) are free Λ_P-modules of finite rank. Since Ext^1_Λ_P(M,N) vanishes if M is a free Λ_P-module, ^r,*_ is isomorphic, as a Λ_P-module, to the direct sum the terms at the two ends of (<ref>).

Now we have established (i) in Theorem <ref>.

For 0≤ r≤ n_d, the _F T_P(_p)-module of Hida families of p-adic Siegel modular forms ordinary for the parabolic P vanishing along the strata associated with cusp labels of rank > r, is defined as 

    M^r_:=Hom_Λ_P(^r,*_,Λ_P).

Similarly, define

    M^0_P_n-r- :=Hom_Λ_P_n-r(^0,*_V,P_n-r- ,Λ_P_n-r).

Applying Hom_Λ_P(·,Λ_P) to (<ref>) gives (iii) of Theorem <ref>.

Let ^P∈_(T_P(_p),^×_p) be an arithmetic dominant weight. Attached to it is a prime ideal _^P of _F T_P(_p). Then unfolding the definitions, one gets the following isomorphisms, 

    .^r_⊗_F T_P(_p)/_^P.∼((^r_[^P])^*,_F)∼_m_l e_PV^SP,r_m,l[^P]

equivariant under the action of the unramfied Hecke algebra away from Np and the ^P_p-operators. Combining (<ref>) with the embedding (<ref>) proves (ii) in Theorem <ref>.



For applications in <ref>, define T^r,N_ as the O_F T_P(Z_p)-algebra generated by all the unramified Hecke operators away from Np∞ and the U_p-operators U^P_p,1, U^P_p,2,…,U^P_p,n acting on ^r_. The algebra T^r,N_ is finite and torsion free over O_F T_P(Z_p). 




§ P-ADIC L-FUNCTIONS

In this section, for a given geometrically irreducible component of Spec(T^0,N_⊗ F), we construct the (d+1)-variable p-adic standard L-function and its d-variable improvement as called in <cit.> (missing the cyclotomic variable). The construction uses the doubling method formula as the integral representation for the standard L-function. The d-variable improvement will be used to employ the Greenberg–Stevens method to prove Theorem <ref> on the derivatives of cyclotomic p-adic L-functions at the so-called semi-stable trivial zeros. The Hida theory for non-cuspidal Siegel modular forms developed in the previous section will be used for the construction of the d-variable improved p-adic L-function.

Before starting the construction, we briefly mention several works on constructing p-adic L-functions using the doubling method. It is Böcherer and Schmidt <cit.> who first carried out such a construction in the special case where π is fixed and is (n)-ordinary with π_∞ isomorphic to a scalar weight holomorphic discrete series. Later, the case where π varies in a cuspidal Hida family which is ordinary for the Borel subgroup is treated in <cit.> for symplectic groups and in <cit.> for unitary groups. Here we look at the more general case of P-ordinary Hida families for a general parabolic P. Moreover, we also construct its improvement as an important input for applying the Greenberg–Stevens method. We also compute the archimdean zeta integrals by using <cit.> and the archimedean functional equations. For groups of general rank, other than the doubling method, there are works on the construction of p-adic L-functions for (n)×(n-1) <cit.> and for (2n) <cit.>, but in their paper, the local zeta integrals at p and ∞ have not been fully computed to verify completely the conjectured interpolation properties of the constructed p-adic L-functions. 




 §.§ Generalities on standard L-functions for symplectic groups

Let π⊂A_0(G(Q)\ G(A)) be an irreducible cuspidal automorphic representation of G(A) and ξ:Q^×\A^×C^× be a finite order Dirichlet character. Take S to be a finite set of places of Q containing the archimedean place and all the finite places where π_v or ξ_v is ramified. 

For v∉ S, there exist unramified characters θ_i:Q_v^×C^×, 1≤ i≤ n, such that π_v is isomorphic to the normalized induction Ind_B_G(Q_v)^G(Q_v)(θ_1,…,θ_n) as G(Q_v)-representations. Put α_v,i=θ_i(q_v) where q_v is the cardinality of the residue field of Q_v. Then α^± 1_v,1,…,α^± 1_v,n are the Satake parameters of π_v, and the unramified local L-factor (for the standard representation ^LG^∘=SO(2n+1,C)GL(2n+1,C)) is defined as

    L_v(s,π_v×ξ_v)=(1-ξ_v(q_v)q_v^-s)^-1∏_i=1^n(1-ξ_v(q_v)α_v,iq_v^-s)^-1(1-ξ_v(q_v)α_v,i^-1q_v^-s)^-1.

The analytic properties (meromorphic continuation, functional equation, location of possible poles) of the partial standard L-function

    L^S(s,π×ξ)=∏_v∉ S L_v(s,π_v×ξ_v)

are established in <cit.>.

Assuming π_∞≅D_, the holomorphic discrete series of weight =(t_1,…,t_n) (so t_1≥⋯≥ t_n≥ n+1), the critical points of L^S(s,π×ξ) are integers s_0 such that

    1≤ s_0≤ t_n-n, (-1)^s_0+n=ξ(-1), or  n+1-t_n≤ s_0≤ 0,  (-1)^s_0+n+1=ξ(-1).

The algebracity of these critical L-values divided by certain automorphic periods (expressed in terms of Petersson inner product) is obtained in <cit.>.  



 §.§ The doubling method for symplectic groups

One standard way to study the standard L-function L^S(s,π×ξ) and its critical values is to apply the doubling method developed by Piatetski-Shapiro–Rallis <cit.>, Garrett <cit.> and Shimura <cit.>. 

For the convenience of the reader, we briefly recall the setting for the doubling method used in <cit.>. Let V' be another copy of V with standard basis e'_1,…,e'_n,f'_1,…,f'_n. Put W=V⊕V', for which we fix the basis e_1,…, e_n,e'_1,…,e'_n,f_1,…,f_n,f'_1,…,f'_n. Then W is endowed with a symplectic pairing induced from that of V and V'. Let H=()=Sp(4n). There is the (holomorphic) embedding ι of G× G into H given by

    ι:G× G    H
    ([ a b; c d ])×([ a' b'; c' d' ])   ⟼([  a  0  b  0;  0 a'  0 b';  c  0  d  0;  0 c'  0 d' ]).


The space W=∑_i=1^nZe_i+Ze'_i is a maximal isotropic subspace of . Its stabilizer Q_H is the standard Siegel parabolic subgroup of H. Besides W, there is another maximal isotropic subspace relevant to us, which is 
W^={(v,(v)):v∈ V}, where : is the involution given by the matrix [   0 I_n; I_n   0 ] with respect to our fixed basis. Note that  dose not preserve the symplectic pairing but has similitude -1. The space W^ is spanned by e_i+f'_i,f_i+e'_i, 1≤ i≤ n. The doubling Siegel parabolic Q^_H is defined to be the stabilizer of W^. We have

    Q^_H=S Q_H S^-1   with   S=[ I_n   0   0   0;   0 I_n   0   0;   0 I_n I_n   0; I_n   0   0 I_n ].
 

For an element g∈ G, define g^ to be g∈ G. This conjugation by  is called the MVW involution. The MVW involution of an irreducible smooth representation of G(Q_v) is isomorphic to its contragredient <cit.>. For φ∈π we define its MVW involution φ^ as φ^(g)=φ(g^). Thanks to the multiplicity one theorem <cit.>, φ^ lies inside π⊂A_0(G(Q)\ G(A)).


Our formulation here aligns with those in <cit.> but differs from <cit.> in that the embedding used in <cit.> corresponds to the above defined ι composite with a conjugation by ϑ on the second copy of G. Hence in our later computation using formulas from <cit.>, the involution ϑ shows up a lot.


Let s be a complex variable. Denote by ξ_s (resp. ξ^_s) the character of Q_H(A) (resp. Q^_H(A)) sending [    A    B;    0 A^-1 ] (resp. S[    A    B;    0 A^-1 ]S^-1) to ξ(det A)|det A|^s. Let I_Q_H(s,ξ)=Ind_Q_H(A)^H(A)ξ_s (resp. I_Q^_H(s,ξ)=Ind_Q^_H(A)^H(A)ξ^_s) be the normalized induction consisting of smooth functions f on H(A) that satisfy f(qh)=ξ_s(q)δ^1/2_Q_H(q)f(h) (resp. f(qh)=ξ^_s(q)δ^1/2_Q^_H(q)f(h)) for all h∈ H(A) and q∈ Q_H(A) (resp. q∈ Q^_H(A)). Recall that the modulus character δ_Q_H takes value |det A|^2n+1/2 at [    A    B;    0 A^-1 ]. The local degenerate principal series I_Q_H,v(s,ξ), I_Q^_H,v(s,ξ) for all places of Q are defined similarly. There is the simple isomorphism

    I_Q_H(s,ξ)    I_Q^_H(s,ξ)
    
       f(s,ξ)   ⟼ f^(s,ξ)(h)=f(s,ξ)(S^-1h).


Given f(s,ξ)∈ I_Q_H(s,ξ), the associated Siegel Eisenstein series is defined as 

    E(h,f(s,ξ))=∑_γ∈ Q_H(Q)\ H(Q) f(s,ξ)(γ h)=∑_γ∈ Q^_H(Q)\ H(Q) f^(s,ξ)(γ h).

This sum is absolutely convergent for Re(s)≫ 0 and admits a meromorphic continuation. 

For a finite place v we fix the Haar measure on G(Q_v) such that the maximal compact subgroup G(_v) has volume 1. For the archimedean place we fix for G(R) the product measure of the one on the maximal compact subgroup K_G,∞={[  a  b; -b  a ]:a+bi∈ U(n,R)} which has total volume 1 with the one on G(R)/K_G,∞=H_n={z∈(n,C):Im z>0} given by det(y)^-n-1∏_1≤ i≤ j≤ ndx_ij dy_ij. The Haar measures on G(A) is taken to be the product of the local ones. 

For a given irreducible cuspidal automorphic representation π⊂A_0(G(Q)\ G(A)) and its complex conjugation π⊂A_0(G(Q)\ G(A)), which is isomorphic to the contragredient of π, we fix isomorphisms π≅⊗'_vπ_v and π≅⊗'_vπ̃_v such that for factorizable φ_1,φ_2∈π with images ⊗_vφ_1,v∈⊗'_vπ̃_v and ⊗_vφ_2,v∈⊗'_vπ_v, we have

    <φ_1,φ_2>=∏_v<φ_1,v,φ_2,v>_v,

where the pairing on the left hand side is the bi-C-linear Petersson inner product with respect to our fixed Haar measure on G(A) and the pairing on the right hand side is the natural pairing between π_v and its contragredient π̃_v. 

For a local section f_v(s,ξ)∈ I_Q_H,v(s,ξ), define

    T_f_v(s,ξ):π   π
    φ   ⟼(T_f_v(s,ξ)φ)(g)=∫_G(Q_v)f^_v(s,ξ)(ι(g'_v,1))φ(gg'_v)d_vg'_v.

We need to be careful with the convergence issue here, especially for v=p,∞. The doubling local zeta integral is defined as

    Z_v(f_v(s,ξ),·,·):π̃_v×π_v   C
    
       (ṽ_1,v_2)   ⟼ Z_v(f_v(s,ξ),ṽ_1,v_2)=∫_G(Q_v)f^_v(s,ξ)(ι(g_v,1))<π̃_v(g_v)ṽ_1,v_2>_vd_vg_v.

For factorizable φ_1,φ_2∈π, we have

    <T_f_v(s,ξ)φ_1,φ_2>=Z_v(f_v(s,ξ),φ_1,v,φ_2,v)/<φ_1,v, φ_2,v>_v<φ_1,φ_2>.


Given φ∈π, we define the linear form

    _φ:A(G(Q)× G()\ G(A)× G())   A(G(Q)\ G(A))
    
       F   ⟼_φ(F)(g)=∫_G(Q)\ G(A)F(g',g)φ(g') dg'.

The doubling method formula is a formula on

    _φ(E(·,f(s,ξ))|_G× G),

involving the partial standard L-function of π and some local zeta integrals.

For a finite place v where ξ is unramified, we denote by f^ur_v(s,ξ) the unique section in I_Q_H,v(s,ξ) that is fixed by the maximal compact subgroup H(_v)⊂ H(Q_v) and takes value 1 at the identity. 
[<cit.>]
Suppose f(s,ξ)=⊗_s∉ Sf^ur_v(s,ξ)⊗⊗_v∈ Sf_v(s,ξ) is a section inside to I_Q_H(s,ξ).  If φ∈π^K^S_G with K^S_G=∏_v∉ SG(_v), then

    _φ(E(·,f(s,ξ))|_G× G)=d^S(s,ξ)^-1· L^S(s+1/2,π×ξ)·(∏_v∈ ST_f_v(s,ξ)φ)^.

Equivalently for all factorizable φ_1,φ_2∈π^K^S_G,

    <E(·,f(s,ξ))|_G× G, φ_1⊗φ^_2>=d^S(s,ξ)^-1· L^S(s+1/2,π×ξ)·∏_v∈ SZ_v(f_v(s,ξ),φ_1,v,φ_2,v)/<φ_1,v, φ_2,v>_v<φ_1,φ_2>.

Here d^S(s,ξ)=∏_v∉ Sd_v(s,ξ) with

    d_v(s,ξ):=L_v(s+2n+1/2,ξ)∏_j=1^nL_v(2s+2n+1-2j,ξ^2).


For later use we also define the normalized Siegel Eisenstein series

    E^*(h,f(s,ξ))=d^S(s,ξ)E(h,f(s,ξ)).

Then the identity (<ref>) from the above theorem becomes

    _φ(E^*(·,f(s,ξ))|_G× G)= L^S(s+1/2,π×ξ)·(∏_v∈ ST_f_v(s,ξ)φ)^.


The identities provided by the doubling method reduce the study of the standard L-function L^S(s,π×ξ) to that of the Siegel Eisenstein series E(·,f(s,ξ)), or more precisely its restriction to G× G, and local zeta integrals at places v∈ S. 



 §.§ The modified Euler factor at p

Before starting the construction of p-adic L-functions, we first recall some basic theory of Jacquet modules and unfold the definition in <cit.> in our case to write down explicitly the expected modified Euler factor at p in the interpolation formula. We also define the modified Euler factor at p for the improved p-adic L-function, and see that when restricting to the leftmost critical points with χ=ϵ^P_d, the difference of the two factors lies inside a finite extension of _F[[T_P(_p)^∘]].



  §.§.§ Jacquet modules and Up-operators

Suppose π_p is the component at the place p of an irreducible automorphic representation π generated by a P-ordinary Siegel modular form. Our discussion on Jacquet modules aims to: (1) show the uniqueness of the P-ordinary forms inside π, or more precisely that the space of P-ordinary Siegel modular forms projects into a one dimensional subspace inside π_p; (2) explain how to retrieve the information on π_p from the eigenvalues of the ^P_p-operators. The uniqueness result will also play an important role in our later computation of the local zeta integral at p. 

Let P_G (resp. SP_G, U_P_G) be the inverse image of P (resp. SP, U_P) of the projection (<ref>). The Jacquet module of π_p associated to the parabolic P_G is defined as

    J_P_G(π_p)=.V_π_p/{π_p(u)· v-v:u∈ U_P_G(Q_p),  v∈ V_π_p}..

It follows from Jacquet's lemma <cit.> that J_P_G(π_p) is naturally isomorphic to the following subspace of V_π_p, 

    ⋂_=(a_1,…,a_1_n_1,…,a_d,…,a_d_n_d)
    a_1≥⋯≥ a_d≥ 0{∫_U_P_G(Q_p)π_p(up^)· v du:v∈ V_π_p},

where p^=diag(p^a_1,…,p^a_1,…,p^a_d,…,p^a_d,p^-a_1,…,p^-a_1,…,p^-a_d,…,p^-a_d). Denote by M_P the Levi subgroup of P and we identify it with the Levi subgroup of P_G via (<ref>). Both J_P_G(π_p) and the space in (<ref>) are equipped with a natural action of M_P(Q_p), and the isomorphism between them is M_P(Q_p)-equivariant. 

Given irreducible smooth admissible representations σ_i of GL(n_i,Q_p), 1≤ i≤ d, Frobenius reciprocity gives

    Hom_G(Q_p)(π_p,Ind_P_G(Q_p)^G(Q_p)σ_1⊠σ_2⊠⋯⊠σ_d)≅Hom_M_P(Q_p)(J_P_G(π_p),(σ_1⊠σ_2⊠⋯⊠σ_d)⊗δ^1/2_P_G),

where δ_P_G is the modulus character sending diag(b_1,b_2,…,b_d)∈ M_P(Q_p), b_i∈(n_i,_p), to ∏_i=1^d|det(b_i)|_p^2n+1+n_i-2N_i.

Suppose that the P-ordinary Siegel modular form generating π is of weight =(^P) with t^P_1≥…≥ t^P_d≥ n+1, so π_∞≅D_(^P). Denote by π_(^P) the subspace of π consisting of forms whose projection to π_∞ belongs to the lowest K_∞-type in D_(^P). Let X^Σ_Γ∩Γ_SP(p^l) be a toroidal compactification of the Shimura variety of level Γ∩Γ_SP(p^l). There is the canonical embedding

    H^0(X^Σ_Γ∩Γ_SP(p^l), ω_(^P))  M_(^P)(_n,Γ∩Γ_SP(p^l))φ_G(·,e_can)A(G(Q)\ G(A)/Γ∩Γ_SP(p^l))_(^P)

from Siegel modular forms defined as global sections of the automorphic sheaf ω_ into automorphic forms on G(A) of  K_∞-type =(^P) (see, for example, <cit.> for precise definition of this embedding). Here for a congruence subgroup Γ, we denote by Γ the corresponding compact subgroup of G(A).

Under the embedding (<ref>), the U^P_p-operator U^P_p,=∏_i=1^n( U^P_p,i)^a_i, for =(a_1,a_2,…,a_n) with a_1≥ a_2≥…≥ a_n≥ 0, on the left hand side corresponds to the following operator on the right hand side,

    U^P_p,=p^<+2ρ_G,c, >∫_SP_G(Z_p)π_p(up^) du,

where ρ_G,c=(n-1/2,n-3/2,…,1-n/2) is the half sum of positive compact roots of G.

We have assumed that π contains a P-ordinary Siegel modular. It follows immediately from the definition of the P-ordinarity and (<ref>), (<ref>) that

    J_P_G(π_p)^SL(n_1,Z_p)×…×SL(n_d,Z_p)≠{0},

which combined with the Frobenius reciprocity (<ref>) implies that there exists spherical representations σ_i of GL(n_i,Q_p), 1≤ i≤ d, and continuous characters η_1,…,η_d of Q_p^× taking value 1 at p, such that 

    π_pInd_P_G(Q_p)^G(Q_p)(σ_1⊗η_1∘det)⊠(σ_2⊗η_2∘det)⊠⋯⊠(σ_d⊗η_d∘det).

In particular, π_p embeds into a principal series. In general, π_p being isomorphic a subquotient of a principal series is equivalent to π containing a finite slope form. 
   

It does not make sense to say P-ordinarity for a purely local representation π_p as the normalization in the definition of the U^P_p-operators depends on the weight of the holomorphic discrete series at the archimedean space. However, being of finite slope is a purely local property.


Next, we say more about the relation between the Satake parameters of the σ_i's in (<ref>) and the eigenvalues of the U^P_p-operators. Let =(θ_1,…,θ_n) be an n-tuple of  continuous characters of Q_p^× (valued in C^×), viewed as a character of T_G(Q_p) via our fixed isomorphism of G^n_m with T_G, such that π_p is isomorphic to a subquotient of the principal series Ind_B_G(Q_p)^G(Q_p). We consider the eigenvalues for the ^P_p-action on Ind_B_G(Q_p)^G(Q_p).

Denote by W_G (resp. W_P_G) the Weyl group with respect to T_G (resp. the subgroup of W_G that maps P_G to itself). Define [W_G/W_P_G] to be the subset of W_G consisting of representatives of smallest lengths of elements in W_G/W_P_G. An element w∈ W_G acts on  by sending it to ^w(t)=(w^-1tw), t∈ T_G. Like , via our fixed isomorphism between T_G and ^n_m, we can write ^w as an n-tuple of characters (θ^w_1,…,θ^w_n).

It follows from <cit.> that the M_P(_p)-representation J_P_G(Ind_P_G(Q_p)^G(Q_p)) has a filtration with graded pieces as

    _i=1^d Ind^GL(n_i,Q_p)_B_n_i(Q_p)(θ^w_N_i-1+1,…,θ^w_N_i)·δ_B_n_i^-1/2.δ_B_G^1/2|_B_n_i,    w∈ [W_G/W_P_G],

where B_n_i is the standard Borel subgroup of GL(n_i) with modulus character δ_B_n_i.

Thus, the dimension of the SL(n_1,Z_p)×…×SL(n_d,Z_p)-invariant space inside J_P_G(π_p) is at most |W_G/W_P_G|=2^n·|S_n/(S_n_1×⋯×S_n_d)|. Each w∈ [W_G/W_P_G] corresponds to an eigensystem of the ^P_p-operators, and the existence of a P-ordinary Siegel modular form in π indicates that there exists w∈ W_G satisfying

    ∑_j=1^r v_p(θ^w_N_i-1+j(p))   ≥ -r(t^P_i-N_i-1-r+1/2),    1≤ r≤ n_i, 1≤ i≤ d, 
    ∑_j=1^n_iv_p(θ^w_N_i-1+j(p))   =-n_i(t^P_i-N_i-1+N_i+1/2)    1≤ i≤ d.

These conditions on the p-adic valuation of θ^w_i, 1≤ i≤ n, imply

    -(t^P_i-N_i-1+1)≤ v_p(θ^w_N_i-1+1(p)),…,v_p(θ^w_N_i(p))≤ -(t^P_i-N_i),    1≤ i≤ d.
 
It is easily seen that given , there is at most one w∈ [W_G/W_P_G] to make (<ref>) hold. By rearranging the θ^± 1_i's, we can assume that w=1 in (<ref>), (<ref>), (<ref>), and that v_p(θ_1(p))≤⋯≤ v_p(θ_n(p))≤ 0. 

The above discussion proves the following proposition.

Suppose that π is an irreducible automorphic representation of G(A) containing a nonzero P-ordinary holomorphic Siegel modular form of weight (^P), t^P_d≥ n+1, and p-nebentypus ^P. 


  * There exists unramified characters θ_1,…,θ_n:^×_p^× satisfying 

    |_^×_p   =(ϵ^P-1_1,…,ϵ^P-1_1_n_1,…,ϵ^P-1_d,…,ϵ^P-1_d_n_d),
    (p)   =(α_1,α_2,…,α_n),    …≤ -(t^P_i-N_i-1+1)≤ v_p(α_N_i-1+1)≤…≤ v_p(α_N_i)
          ≤ -(t^P_i-N_i)≤-(t_i+1^P-N_i+1)≤ v_p(α_N_i+1)≤…
 
such that π_pInd^G(_p)_B_G(_p).

  * Let a_i be the eigenvalue for the action of U^P_p,i on the P-ordinary form in π. Then a_N_1,…,a_N_d are p-adic units given by

    a_N_i=∏_j=1^i p^n_j(t^P_j-N_j-1+N_j+1/2)α_N_j-1+1α_N_j-1+2⋯α_N_j,

More generally, for N_j≤ i≤ N_j+1, the eigenvalue a_i is an p-adic integer given as 

    a_i=a_N_j· p^(i-N_j)(t^P_i+1-N_j+i+1/2)∑_ϱ∈S_n_j+1/S_i-N_j×S_N_j+1-iα_N_j+ϱ(1)α_N_j+ϱ(2)⋯α_N_j+ϱ(r).


  * Inside Ind^G(_p)_B_G(_p), there is a unique generalized eigenvector (up to scalar) for the operator U^P_p=∏_i=1^d U^P_p,N_i with eigenvalue being a p-adic unit. In particular, under the projection ππ_p, the image of P-ordinary Siegel modular forms is one dimensional.




Let  be as in the above proposition. If ϵ^P_d=triv and α_n=p^-1, since the P-ordinary condition implies that π_pInd^G(_p)_B_G(_p), in the Weil–Deligne representation attached to π_p, there should be a nontrivial monodromy between the eigenspaces with Frobenius  eigenvalues 1 and α_n=p^-1.




  §.§.§ The modified Euler factor at p for p-adic interpolation

If we consider the Weil–Deligne representation attached to π_p, the eigenvalues of Frobenius are 1, α^± 1_1,…,α^± 1_n. Meanwhile, for the p-adic representation associated to π <cit.>, the Hodge–Tate weights are 0,± (t^P_1-1),…,± (t^P_1-n_1),…,± (t^P_d-(N_d-1+1)),…,± (t^P_d-n). Thus, (<ref>) and (<ref>) essentially say that the Newton polygon is above the Hodge polygon and the two polygons meet at the points with horizontal coordinates 0, N_1, N_2,…,N_d,2n+1-N_d,…,2n+1-N_2,2n+1-N_1,2n+1.

Since the definition of the modified Euler factor in <cit.>, formulated in terms of the Weil–Deligne representation, does not depend on the monodromy operator, our above description of the Weil–Deligne representation associated to π_p is enough for us to unfold the definition in this case to obtain the explicit modified Euler factor at p in terms of the ^P_p-eigenvalues.

From now on we fix a (tame) finite order Dirichlet character ϕ:Q^×\A^×C^× unramified away from N∞. Suppose χ∈Hom_cont(Z^×_p,Q^×_p) is of finite order. We also view it also as a C^×-valued character of Q^×\A^× which sends the uniformizer in _v to χ(q_v) for finite places v≠ p. In the same way, we view the finite order characters ϵ^P_1,…,ϵ^P_d as adelic characters. Let  be as in Proposition <ref>. Denote by σ_i the unramified representation of (n_i,_p) such that σ_i⊗ϵ^P_i,p∘ =Ind^(n_i,_p)_B_n_i(_p)(θ_N_i-1+1,…,θ_N_i). Denote by σ_i the contragredient of σ_i.

The modified Euler factor at p for p-adically interpolating the critical values of L^S(s,π×ϕχ) to the left of the center is

    E_p(s,π×ϕχ)=∏_i=1^d γ_p(s,σ_i⊗ϕ_pχ_pϵ^P-1_i,p)^-1.

Here we omit our fixed additive character _p from the usual notation for gamma factors. One can also write the gamma factors in terms of the Satake parameters as

    γ_p(s,σ_i⊗ϕ_pχ_pϵ^P-1_i,p)^-1=∏_j=N_i-1+1^N_i1-ϕ_p(p)^-1α_j^-1p^s-1/1-ϕ_p(p)α_jp^-s,   if χϵ^P-1_i is trivial,
    
       G(χ^-1ϵ^P_i)^n_i∏_j=N_i-1+1^N_i(ϕ_p(p)^-1α^-1_jp^s-1)^c(χϵ^P-1_i),    otherwise,

where p^c(χϵ^P-1_i) is the conductor of χϵ^P-1_i, and the Gauss sum is defined as

    G(χ^-1ϵ^P_i)=∫_p^-mZ_pχ_pϵ^P-1_i,p(x)_p(-x) dx,   m≫ 0.


We also define the improved modified Euler factor at p for the d-variable improved p-adic L-function. The improved p-adic L-function is supposed to interpolate the leftmost critical L-values with χ=ϵ^P_d. Define

    E^_p(s,π×ϕϵ^P_d)   =∏_i=1^d-1γ_p(1-s,σ_i⊗ϕ^-1_pϵ^P-1_d,pϵ^P_i,p)· L_p(s,σ_d⊗ϕ_p)

It is easy to see that by (<ref>) and (<ref>) both E_p(s,π×ϕχ) and E^_p(s,π×ϕϵ^P_d) can be written in term of the ^P_p-eigenvalues of the P-ordinary Siegel modular form contained in π. 


We have

    E_p(n+1-t^P_d,π×ϕϵ^P_d)=A^P(π×ϕϵ^P_d)· E^_p(n+1-t^P_d,π×ϕϵ^P_d)

with

    A^P(π×ϕϵ^P_d)   =∏_j=N_d-1+1^n(1-ϕ_p(p)^-1α^-1_jp^n-t^P_d)
       = 1+a_n^-1a_N_d-1(-ϕ_p(p)^-1p^n_d-1/2)^n_d+a_n^-1∑_r=1^n_d-1a_N_d-1+r(-ϕ_p(p)^-1p^n_d-1-r/2)^n_d-r.

Since all the a_i's are the U^P_p-eigenvalues of the P-ordinary Siegel modular forms, when (the eigensystem of) π varies in a P-ordinary Hida family, A^P(π,ϕ) becomes a d-variable p-adic analytic function lying inside a finite extension of O_F T_P(Z_p)^∘. This explains why when restricting to the leftmost critical values with χ=ϵ^P_d, one expects the existence of the improved d-variable p-adic L-function with the improved modified Euler factor E^_p(π,ϕ) at p (improved in the sense of saving part of the numerator from E_p(n+1-t^P_d,π×ϕϵ^P_d)).





 §.§ The choices of local sections for the Siegel Eisenstein series

Our choices of local sections for Siegel Eisenstein series on (4n),  and the formulae for local Fourier coefficients as well as the doubling local zeta integrals corresponding to those selected sections are summarized in the two tables in <ref>. This section explains the strategy for the section selections. The computation of the zeta integrals at the place p is done in <ref>.



  §.§.§ Criteria for selecting sections
We first describe the context   and criteria for choosing the sections for Siegel Eisenstein series on (4n). Recall that Hom_cont(T_P(Z_p),Q^×_p) is the weight space for Hida families which are ordinary for the parabolic for P. An arithmetic point in it corresponds to a character ^P=^P_alg·^P_f∈Hom_cont(T_P(Z_p),Q^×_p), a product of the algebraic part ^P_alg=^P=(t^P_1,…,t^P_d) and the finite order part ^P_f=^P=(ϵ^P_1,…,ϵ^P_d). Similarly, the parameterization space for the cyclotomic variable is Hom_cont(Z^×_p,Q^×_p), and an arithmetic point in it is a character κ=κ_alg·κ_f, with algebraic part κ_alg=k and finite order part κ_f=χ. We call an arithmetic point ^P (resp. (κ,^P)) admissible if t^P_1≥…≥ t^P_d≥ n+1 and (resp. t^P_1≥…≥ t^P_d≥ k≥ n+1).

Let _P be a geometrically irreducible component of Spec(T^0,N_⊗_O_FF). The projection of _P to the weight space is one of its |T_P(/p)| connected components. We say the parity of _P is compatible with ϕ if all the points ^P in that connected component satisfy ^P_d(-1)=ϕ(-1).

A point x∈_P(Q_p) is called arithmetic if its projection ^P inside the weight space is arithmetic, and an arithmetic pair (κ,x) (resp. an arithmetic point x) is called admissible if (κ,^P) (resp. ^P) is admissible. If the Hecke eigensystem parametrized by x appears in an irreducible cuspidal automorphic representation π_x∈_0(G()\ G()) with π_x,∞≅_(^P), we call such an x classical, and one can define the corresponding L^Np∞(s,π_x×ϕχ), E_p(s,π_x×ϕχ), E^P-imp_p(π_x,ϕ). Note that because of the lack of strong multiplicity one, π_x may not be unique, but the partial L-functions and the modified Euler factors at p do not depend on the choice of π_x.

The (d+1)-variable p-adic L-function is intended to interpolate the critical values

    E_p(n+1-k,π_x×ϕχ)· L^Np∞(n+1-k,π_x×ϕχ)

divided by a Petersson inner product period for (κ,x) admissible with κ(-1)=ϕ(-1) and x classical (by our construction, if (κ,x) is admissible but x is not classical, one can see that the evaluation of our p-adic L-function there is 0). Its d-variable improvement (assuming the parity of _P is compatible with ϕ) is supposed to interpolate

    E^_p(n+1-t^P_d,π_x×ϕϵ^P_d)· L^Np∞(n+1-t^P_d,π_x×ϕϵ^P_d)

divided by a Petersson inner product period for classical x. 

From Theorem <ref>, we see that in order to get the above L-values, we need to pick a Siegel Eisenstein series on (4n) with nice properties for each admissible (κ,^P)∈Hom_cont(Z^×_p× T_P(Z_p,Q^×_p)) with κ(-1)=ϕ(-1) as well as for each admissible ^P∈Hom_cont(T_P(Z_p,Q^×_p)) with ^P_d(-1)=ϕ(-1), so that we can deduce the desired congruences among the L-values from those of the Siegel Eisenstein series. Picking the Siegel Eisenstein series amounts to selecting sections in the degenerate principal series. 

More precisely, for (κ,^P) (resp. ^P) as above and each place v of , we need to pick a section f_κ,^P,v (resp. f_^P,v)  from I_Q_H,v(2n+1/2-k,ϕχ) (resp. I_Q_H,v(2n+1/2-t^P_d,ϕϵ^P_d)), such that 


  * We have enough control of the local zeta integrals at places dividing Np∞. In particular, we are able guarantee the non-vanishing of the archimedean zeta integrals and compute the zeta integrals at p.

  * The collection of the Eisenstein series E(·,f_κ,^P)|_G× G (resp. E(·,f_^P)|_G× G) (after suitable normalizations)  assembles to a p-adic family.

The way we treat the second requirement is via looking at their Fourier coefficients and invoking the q-expansion principle. Also, the second requirement provides us a hint for making the choices of the sections at p based on our selection of archimedean sections.



  §.§.§ The Fourier coefficients for Siegel Eisenstein series

For ∈Sym(2n,Q), the -th Fourier coefficient of E(·,f(s,ξ)) is defined as

    E_(h,f(s,ξ)):=∫_Sym(2n,Q)\Sym(2n,A) E([ I_2n    ς;    0 I_2n ]h,f(s,ξ))_A(-Trς) dς.

Suppose f(s,ξ)=⊗_v f_v(s,ξ) is factorizable. If ()≠ 0 or there exists a finite place v such that f_v(s,ξ) is supported on the “big cell” Q_H(Q_v)[     0 -I_2n;  I_2n     0 ] Q_H(Q_v), then

    E_(h,f(s,ξ))=∏_v W_,v(h,f(s,ξ))
 
with

    W_,v(h_v,f_v(s,ξ))=∫_Sym(2n,Q_v)f_v(s,ξ)([     0 -I_2n;  I_2n     ς ]h_v)_v(-Trς) d_vς.


For =x+√(-1)y, a point in the Siegel upper half space H_2n, set h_=1_f·[     √(y) x√(y)^-1;        0  √(y)^-1 ]∈ H(A). It is a standard fact that if E(h_,f(s_0,ξ)) is nearly holomorphic as a function in  for some s_0∈ 2^-1·Z, then E_(h_,f(s_0,ξ)) gives the -th coefficient of the q-expansion associated to E(h,f(s_0,ξ)) viewed as a p-adic form by the maps (<ref>)(<ref>).



  §.§.§ The unramified places

For v∤ Np∞, we simply take

    f_κ,^P,v   =.f^ur_v(s,ϕχ)|_s=2n+1/2-k,     f_^P,v   =.f^ur_v(s,ϕϵ^P_d)|_s=2n+1/2-t^P_d.

The formulae for W_(1_v,f_κ,^P,v) and W_(1_v,f_^P,v) are computed by Shimura <cit.> and are listed in the tables in <ref>. The formulae for the local zeta integrals are part of Theorem <ref>.



  §.§.§ The archimedean place

For an integer k≥ n+1 satisfying ξ(-1)=(-1)^k, the classical section of weight k in I_Q_H,∞(s,ξ) is defined as

    f_∞^k(s,sgn^k)(h)=j(h,i)^-k|j(h,i)|^k-(s+2n+1/2)
 
where j(h,i)=(μ(h,i))= (Ci+D) for h=[ A B; C D ]. 

Let μ^+_0=(μ^+_0,ij)_1≤ i,j≤ n, where the entries are elements inside (LieH)_C given as

    μ^+_0,ij=[        I_2n √(-1)· I_2n; √(-1)· I_2n        I_2n ][    0    0    0 E_ij;    0    0 E_ji    0;    0    0    0    0;    0    0    0    0 ][        I_2n √(-1)· I_2n; √(-1)· I_2n        I_2n ]^-1,
 
where E_ij is the n× n matrix with 1 as the (i,j)-entry and 0 elsewhere. 

The μ^+_0,ij's act on A(H(Q)\ H(A)) by differentiating the right translation of H(R). Their realizations on the Siegel upper half space are the Maaß–Shimura differential operators (see <cit.>). 

For admissible (κ,^P) (resp. ^P) with ϕχ(-1)=(-1)^k (resp. ϕϵ^P_d(-1)=(-1)^t^P_d), set

    f_κ,^P,∞(s)=   ∏_i=1^d-1det((μ^+_0)_N_i/4π√(-1))^t^P_i-t^P_i+1det(μ^+_0/4π√(-1))^t^P_d-k f^k_∞(s,sgn^k),
    
       f_^P,∞(s)=   ∏_i=1^d-1det((μ^+_0)_N_i/4π√(-1))^t^P_i-t^P_i+1 f^t^P_d_∞(s,sgn^t^P_d)=f_τ^P_d,^P,∞(s)

and 

    f_κ,^P,∞   =f_κ,^P,∞(2n+1/2-k),    f_^P,∞   =f_^P,∞(2n+1/2-t^P_d)=f_τ^P_d,^P,∞.


The formulae for the corresponding Fourier coefficients (listed in tables in <ref>) are deduced from Shimura's computation <cit.> for the classical scalar weight section and formulae for the action of differential operators on p-adic expansions (see the proof of <cit.>). The proof of the non-vanishing of the corresponding archimedean zeta integrals is postponed to <ref>. 




  §.§.§ The “big cell” section at a finite place

We choose our sections at v|Np from a special type of sections, the so-called “big cell” sections. Given a finite place v and a compactly supported locally constant function α_v on Sym(2n,Q_v), the “big cell” section inside I_Q_H,v(s,ξ) associated to α_v is defined as 

    f_v^α_v(s,ξ)([ A B; C D ])=ξ^-1( C)| C|^-(s+2n+1/2)α_v(C^-1D)    if  C≠ 0,
     0     if  C=0.

An easy computation shows that

    W_,v(1_v,f_v^α_v(s,ξ))= ∫_Sym(2n,_v)α_v(ς)_v(-Trς)  d_vς =α_v().




  §.§.§ The volume sections at places dividing N

For a positive integer N and a place v|N, the volume section f^vol_v(s,ξ) inside I_Q_H,v(s,ξ) is defined as the “big cell” section associated to the characteristic function of the open compact subset

    -[   0 I_n; I_n   0 ]+N(2n,_v)⊂(2n,_v).
 
We set

    f_κ,^P,v   =.f^vol_v(s,ϕχ)|_s=2n+1/2-k,     f_^P,v   =.f^vol_v(s,ϕϵ^P_d)|_s=2n+1/2-t^P_d.


The Fourier coefficients associated to the volume sections are easily computed by computing the Fourier transform of the characteristic function of (<ref>). The computation of local zeta integrals is also easy (the same as <cit.>). See the tables for the formulae.



  §.§.§ The place p

It remains to pick Schwartz functions α_κ,^P and α_^P on (2n,_p), and our choices for f_κ,^P,p (resp. f_^P,p) will the “big cell” section attached to α_κ,^P (resp. α_^P). The criterion for picking them is to make the (p-adic) q-expansions of the resulting Siegel Eisenstein series p-adically interpolable. In fact we will first pick α_κ,^P and α_^P, and then apply inverse Fourier transform to get α_κ,^P and α_^P for computing the local zeta integrals. 

The theory of nearly holomorphic forms and Maaß–Shimura differential operators formulated in terms of automorphic sheaves and their interpretations as p-adic Siegel modular forms are needed in our construction. We will freely use the formulation and notation in <cit.> and <cit.>.

Recall some notation loc. cit.; denote by ^r_ the automorphic bundle of degree r and weight  nearly holomorphic forms over the Siegel variety defined as in <cit.>, and by N^r_(_n,Γ∩Γ_SP(p^l)) the space of vector-valued nearly holomorphic Siegel modular forms on the Siegel upper half space _n of degree r, weight  and level Γ∩Γ_SP(p^l) in the sense of Shimura. There is the embedding

    H^0(X^Σ_G,Γ∩Γ_SP(p^l)× X^Σ_G,Γ∩Γ_SP(p^l),^r_⊠^r_)     N^r_(_n,Γ∩Γ_SP(p^l))⊗ N^r_(_n,Γ∩Γ_SP(p^l))
       φ(·,_can)(G()× G()\ G()× G()).


Generalizing the embedding (<ref>), as explained in <cit.>, the space of nearly holomorphic Siegel modular forms of level Γ∩Γ_SP(p^l) also embeds into the space of p-adic Siegel modular forms,

    H^0(X^Σ_G,Γ∩Γ_SP(p^l)× X^Σ_G,Γ∩Γ_SP(p^l),^r_(^P)⊠^r_(^P))(_m_l V^SP_m,l⊗__FV^SP_m,l)[1/p].


When choosing α_κ,^P and α_^P, we want to ensure that the restrictions to G× G of the resulting adelic Siegel Eisenstein series lie inside the image of the embedding (<ref>), so that we can study them as p-adic Siegel modular forms via (<ref>). Let (2n,_p)^* be the dual space of (2n,_p), which can be identified with (2n,_p) via the Trace pairing, and let (2n,_p)^* be the dual of (2n,_p), which identified with the matrices α in (2n,_p) such that Tr(α a) ∈_p for all a ∈(2n,_p).  Hence we require:


  * α_κ,^P and α_^P take values in a finite extension of .

  * α_κ,^P and α_^P are supported on (2n,_p)^*, and for x∈(2n,_p)^*, a_1,a_2∈ SP(_p),

    α_κ,^P(x)   =α_κ,^P(a_1   0
    0   a_2xa_1   0
    0   a_2),    α_^P(x)   =α_^P(a_1   0
    0   a_2xa_1   0
    0   a_2).



With α_κ,^P and α_^P satisfying these conditions, we can define 

    _κ,^P, _^P   ∈_r_l  H^0(X^Σ_G,Γ∩Γ_SP(p^l)× X^Σ_G,Γ∩Γ_SP(p^l),^r_(^P)⊠^r_(^P)),

as the preimage of the adelic forms

    χ(-1)^nΓ_2n(2n+1/2)/2^2nk-2n^2+nπ^n+2n^2 .E^*(·,f_κ,^P)|_G× G,      ϵ^P_d(-1)^nΓ_2n(2n+1/2)/2^2nt^P_d-2n^2+nπ^n+2n^2·.E^*(·,f_^P)|_G× G.


Here for a positive integer m,

    Γ_m(s):=π^m(m-1)/4∏_j=0^m-1Γ(s-j/2).

In the following we will not distinguish _κ,^P, _^P from their images under the embedding (<ref>).

Set 

    ε_=(_m_l ε^1,1_,m,l, _m_l ε^1,1_,m,l):_m   _l V^SP_m,l⊗__FV^SP_m,l_F N^-1(n,)^∗⊕ 2_≥ 0

with 

    _m_l ε^1,1_,m,l:_m_l V^SP_m,l_F N^-1S^2(X_n)_≥ 0=_F N^-1(n,)^*_≥ 0.

being the p-adic q-expansion map at infinity.

For any element G of _m _l V^SP_m,l⊗__FV^SP_m,l with q-expansion 
    ε_(G) =∑_β_1,β_2∈ N^-1(n,)^*_>0 ∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_>0() q^β_1q^β_2

and any pair (β_1,β_2) of elements in N^-1(n,)^*_≥ 0 we define the (β_1,β_2) Fourier coefficient of G as 

    ε_(β_1,β_2,G ) =∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_>0().


Regarding the q-expansion of _κ,^P and _^P, we have the following proposition.

Let (β_1,β_2) be a pair of elements in N^-1(n,)^*_≥ 0. For admissible (κ,^P)∈_cont(^×_p× T_P(_p),^×_p) with ϕχ(-1)=(-1)^k, we have

    ε_(β_1,β_2,_κ,^P)   =∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_≥ 0_κ,^P(),

where for >0, the coefficient _κ,^P() is given as

    _κ,^P()=   N^-n(2n+1)∏_v|N_v(2Trβ_0) · L^Np∞(n+1-k,ϕχλ_)·∏_v|(2)
    v∤ Np∞ g_,v(ϕχ(q_v)q_v^k-2n-1)
       ×  (-1)^nkχ(-1)^n·α_κ,^P() ∏_i=1^d-1det((2β_0)_N_i)^t^P_i-t^P_i+1det(2β_0)^t^P_d-k.

For admissible ∈_cont(T_P(_p),^×_p) with ϕϵ^P_d(-1)=(-1)^t^P_d, we have 

    ε_(β_1,β_2,_^P)   =∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_≥ 0_^P(),

where for ≥ 0, the coefficient _^P() is given as

    _^P()=   N^-n(2n+1)∏_v|N_v(2Trβ_0) · L^Np∞(n+1-t^P_d-r/2,ϕχλ_)·∏_j=1^corank()/2L^Np∞(2n+3-2t^P_d-2j,(ϕϵ^P_d)^2)
        ×(-1)^nt^P_dϵ^P_d(-1)^n∏_v|^*(2)
    v∤ Np∞ g_,v(ϕχ(q_v)q_v^t^P_d-2n-1)·α_^P() ∏_i=1^d-1det((2β_0)_N_i)^t^P_i-t^P_i+1

if rank() is even, and

    _^P()=   N^-n(2n+1)∏_v|N_v(2Trβ_0) ·∏_j=1^corank()+1/2L^NP∞(2n+3-2t^P_d-2j,(ϕϵ^P_d)^2)
        ×(-1)^nt^P_dϵ^P_d(-1)^n∏_v|^*(2)
    v∤ Np∞ g_,v(ϕχ(q_v)q_v^t^P_d-2n-1)·α_^P() ∏_i=1^d-1det((2β_0)_N_i)^t^P_i-t^P_i+1

if rank() is odd.

Here for ∈(2n,)∩(2n,_v)^*, ^*(2) denotes the product of all the nonzero eigenvalues of 2. If rank() is even, the quadratic character λ_ is defined as λ_(q_v)=((-1)^rank()/2^*()/q_v). The g_,v(·) appearing in above formulae is a polynomial with coefficients in . For an integer m, (2β_0)_m denotes the upper left m× m-minor of 2β_0.


The proof is similar as <cit.>. It relies on the formulae for local Fourier coefficients as listed in the two tables in <ref>, and uses formulae of differential operators on p-adic q-expansions. 


It is not difficult to observe that all the terms in the above formulae for _κ,^P(), _^P() are ready for p-adic interpolation with respect to (κ,^P), ^P, except the last terms

    α_κ,^P() ∏_i=1^d-1det((2β_0)_N_i)^t^P_i-t^P_i+1det(2β_0)^t^P_d-k,
       α_^P()∏_i=1^d-1det((2β_0)_N_i)^t^P_i-t^P_i+1.

(The p-adic interpolation of the Dirichlet L-values in the formulae follows from the existence of the Kubota–Leopoldt p-adic L-function <cit.>.)

In order to make (<ref>) and (<ref>) p-adically interpolable, one needs to require that if  belongs to the support of the Schwartz function α_κ,^P (resp. α_^P), then ((2β_0)_N_i) is a p-adic unit for 1≤ i≤ d (resp. 1≤ i≤ d-1). The very natural choices of α_κ,^P and α_^P are

    α_κ,^P() =   1_p^2(n,_p)^*(β_1)1_(n,_p)^*(β_2)∏_i=1^d1_(N_i,_p)((2β_0)_N_i)
       × ∏_i=1^d-1ϵ^P_iϵ^P-1_i+1(det((2β_0)_N_i))·ϵ^P_dχ^-1(det(2β_0)),

and

    α_^P() =   1_p^2(n,_p)^*(β_1)1_(n,_p)^*(β_2)1_M_n(_p)(β_0)∏_i=1^d-11_(N_i,_p)((2β_0)_N_i)
       × ∏_i=1^d-1ϵ^P_iϵ^P-1_i+1(det((2β_0)_N_i)).

Here M_n denotes the space of n× n matrices.

Since W_,p(1_p,f_κ,^P,p)=α_κ,^P(), our choice (<ref>) makes _κ,^P() vanish unless  is invertible. The semi-positivity of  implies that both β_1 and β_2 are positive definite. Thus ε_(β_1,β_2,_κ,^P) is nonzero only if β_1,β_2>0. Similarly, ε_(β_1,β_2,_^P) is nonzero only if β_1,β_2≥ 0 and their ranks are at least n-n_d.

With α_κ,^P, α_^P being set as in (<ref>), (<ref>), we have

    ε_(_κ,^P)   =∑_β_1,β_2∈ N^-1(n,)^*_>0 ∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_>0_κ,^P() q^β_1q^β_2,
    ε_(_^P)   =∑_β_1,β_2∈(n,)^*_≥ 0
    rk(β_1),rk(β_2)≥ n-n_d ∑_=β_1   β_0
    β_0   β_2∈ N^-1(2n,)^*_≥ 0_^P() q^β_1q^β_2,

and each _κ,^P() (resp. _^P()) appearing here admits p-adic interpolation with respect to (κ,^P) (resp. ^P).

If we look at the q-expansions of _κ,^P and _^P at other cusps in the ordinary locus (which is equivalent to look at W_(ι(g_1,g_2),f_κ,^P), W_(ι(g_1,g_2),f_^P) for g_i∈ G() with g_i,p=1, g_i,∞=g_z_i, i=1,2), the support of α_κ,^P (resp. α_^P) again makes the term indexed by degenerate (β_1,β_2) (resp. β_1 or β_2 of rank <n-n_d) vanish. Hence

    _κ,^P   ∈_m_l V^SP,0_m,l⊗__F V^SP,0_m,l,    _^P   ∈_m_l V^SP,n_d_m,l⊗__F V^SP,n_d_m,l.



Compared to α_κ,^P, the support of α_^P is enlarged. As the cyclotomic variable κ is fixed to be equal to ^P_d, the term (2β_0) does not appear in (<ref>), and one does not need to require 2β_0∈(n,_p) for the support of α_^P(). Later, we will see that it is this relaxation on the support that saves us the factor ^P(π×ϕϵ^P_d) in the local zeta integral for f^α_^P_p(2n+1/2-t^P_d,ϕϵ^P_d) compared to that for .f^α_κ,^P_p(2n+1/2-k,ϕχ)|_κ=^P_d. This relaxation also means that the resulting _^P is not necessarily cuspidal as p-adic forms. Thus, the Hida theory for non-cuspidal Siegel modular forms developed in <ref> is needed to construct the improved p-adic L-function from the _^P's.




  §.§.§ The two tables

In the tables on the next two pages, we summarize our choices of sections for Siegel Eisenstein series, the formulae of the corresponding local Fourier coefficients, and the local zeta integrals.

We explain some notation. In the tables, φ∈π is a P-ordinary cuspidal holomorphic Siegel modular form  of weight (^P) fixed by Γ∩ SP_G(Z_p) with p-nebentypus ^P. 

The operator  is defined as

    (φ)(g)=∫_SP_G(Z_p)φ^(gu) du.

As observed before Remark <ref>, the form φ^ belongs to π and so (φ) belongs to π too. If φ is holomorphic of weight ^P, then so is (φ). The operator  should be viewed as an analogue of the operator sending a modular form f of level Γ_0(N) to f^c|[  0 -1;  N  0 ].. In the same way as <cit.>, one can show that the P-ordinary projection e_P(φ) is nonzero if e_Pφ is nonzero. 

We denote by v_(^P)∈_(^P) the highest weight vector inside the lowest K_G,∞-type, and v^∨_(^P)∈_(^P) is the dual vector to v_(^P).

















 §.§ The construction of mu E and E-imp


Assume ϕ^2≠triv. Then it follows from our choices of the sections f_κ,^P∈ I_Q_H(2n+1/2-k,ϕχ) and f_^P∈ I_Q_H(2n+1/2-t^P_d,ϕϵ^P_d) that there exist p-adic measures 

    μ_,   ∈ eas(^×_p× T_P(_p),_F  N^-1(n,)^*⊕ 2_>0),
    μ_^imp,   ∈ eas (T_P(_p),_F N^-1(n,)^*⊕ 2_≥ 0)

satisfying the interpolation properties

    ∫__p^×× T_P(_p)(κ,^P) dμ_,   ={[                                    ε_(_κ,^P) if (κ,^P)  is admissible with ϕχ(-1)=(-1)^k,;                                            0                              if ϕ(-1)≠κ(-1), ].
    ∫_T_P(_p)^P dμ_^imp,   ={[                                          ε_(_^P) if ^P  is admissible with ϕϵ^P_d(-1)=(-1)^t^P_d,;                                                0                               if ϕ(-1)≠^P_d(-1), ].

(see <cit.> for more details on how to obtain these p-adic measures from the formulae for ε_(_κ,^P), ε_(_^P) in <ref>).




The assumption ϕ^2≠triv is not essential. Without it,  due to the pole of the Kubota–Leopoldt p-adic L-function, we need to make some modification accordingly to allow a possible pole in the constructed measures.


Let V^SP,r,Δ be the subspace of _m_l V^SP,r_m,l⊗__F/p^m V^SP,r_m,l consisting of elements annihilated by γ⊗ 1-1⊗γ for all γ∈ P(_p). By definition and (<ref>), we know that _κ,^P∈ V^SP,0,Δ and _^P∈ V^SP,n_d,Δ. Then due to the Zariski density of the admissible points (κ,^P) (resp. ^P) inside ^×_p× T_P(_p) (resp. T_P(_p)), the measure μ_, (resp. μ_^imp,) lies inside the image of the following embedding induced by p-adic q-expansion

    eas(^×_p× T_P(_p),V^SP,0,Δ)    eas(^×_p× T_P(_p),_F N^-1(n,)^*⊕ 2_>0)
    
       (resp.  eas(T_P(_p),V^SP,n_d,Δ)    eas(T_P(_p),_F N^-1(n,)^*⊕ 2_≥ 0)).

Viewing  μ_, (resp. μ_^imp,) as a p-adic measure valued in V^SP,0,Δ (resp. V^SP,n_d,Δ) , Propositions <ref> and <ref> show that one can apply e_P× e_P to it and get

    μ_,∈ eas(^×_p× T_P(_p),e_PV^SP,0,Δ)  (resp. μ_^imp,∈ eas(T_P(_p),e_PV^SP,n_d,Δ)).

For ∈_(T_P(/p),μ_p-1) and an _F T_P(_p)-module, we use a subscript _ to denote its -isotypic part for the action of T_P(/p). Then like <cit.>,  for all 0≤ r≤ n_d and ^P such that ^P|_T_P(/p)=, we have the commutative diagram 

    eas(T_P(_p),e_PV^SP,r,Δ)^♮_[r]^Φ^Δ_[dr]_μ↦∫_T_P(_p)^P dμ       ^r_,⊗_Λ_P^r_,[d]^_^P
       _m_l e_PV^SP,r_m,l[^P]⊗__Fe_PV^SP,r_m,l[^P],

where eas(T_P(_p),e_PV^SP,r,Δ)^♮ is the subspace of  eas(T_P(_p),e_PV^SP,r,Δ) consisting of measures μ satisfying

    ∫_T_P(_p)^P dμ∈ e_PV^SP,r,Δ[^P],
 
where e_PV^SP,r,Δ[^P] denotes the subspace where T_P(_p) acts via ^P.

Set

    Φ^Δ:=⊕_Φ^Δ_:    eas(T_P(_p),e_PV^SP,r,Δ)^♮=⊕_ eas(T_P(_p),e_PV^SP,r,Δ)^♮_
         ^r_⊗__F T_P(_p)^r_=⊕_^r_,⊗_Λ_P^r_,,

where  runs over _(T_P(/p),μ_p-1). This Λ_P-module morphism Φ^Δ also induces

    Φ^Δ: eas(^×_p× T_P(_p),e_PV^SP,r,Δ)^♮ eas(^×_p,^r_⊗__F T_P(_p)^r_).

As checked before Remark <ref>, the measure μ_, (resp. μ_^imp,) belongs to the domain of Φ^Δ  with r=0 (resp. r=n_d). Hence we can define (with a small abuse of notation)

    μ_,   =Φ^Δ(μ_,)∈ eas(^×_p,^0_⊗__F T_P(_p)^0_),
    ^imp_   =Φ^Δ(μ_^imp,)∈^n_d_⊗__F T_P(_p)^n_d_.

Let ^n_d,cl_ denote the sub-Λ_P-module of ^n_d_ consisting of P-ordinary families whose specializations are classical Siegel modular forms at ^P with t^P_1≫ t^P_2≫⋯≫ t^P_d≫ 0. It follows from our construction that the P-ordinary family ^imp_ is contained in ^n_d,cl_⊗__F T_P(_p)^n_d,cl_ 




 §.§ The p-adic L-functions and their interpolation properties

Let ^0,N_ (resp. ^n_d,N_) be the _F T_P(_p)-algebra as defined at the end of <ref>. Let _P be a geometrically irreducible component of Spec(^0,N_). Because of the natural quotient map ^n_d,N_^0,N_, _P can also be viewed as a geometrically irreducible component of Spec(^n_d,N_). Denote by F__P the function field of _P and by __P the integral closure of Λ_P inside F__P. Let λ__P:^n_d,N___P be the homomorphism of Λ_P-algebra associated to _P.

As F__P-algebras, we have the decomposition

    ^n_d,N_⊗_Λ_P F__P=F__P⊕ R__P

such that the projection onto F__P is given by λ__P. Define 1__P as the element in ^n_d,N_⊗_Λ_P F__P which corresponds to (id,0) in F__P⊕ R__P. 




    1__P(^n_d,cl_)⊂^0_⊗__F T_P(_p) F__P.
   


The cuspidality condition is about the vanishing of the restriction to the boundary, therefore an element in ^n_d_ is cuspidal as long as its specializations at a Zariski dense subset in _(T_P(_p),_p^×) are cuspidal. For all t^P_1≫ t^P_2≫…≫ t^P_d≫ 0, the specialization of ^n_d,cl_ at the algebraic point ^P∈_(T_P(_p),_p^×) consists of classical holomorphic Siegel modular forms of weight (^P) and level Γ(N)∩Γ_P(p). We reduce to show that for all t^P_1≥ t^P_2≥…≥ t^P_d≫ 0, if a Hecke eigenform φ (for all unramified Hecke operators away from Np∞) in M^n_d_(^P)(Γ(N)∩Γ_P(p);) shares the same eigenvalues as a cuspidal Hecke eigenform in M^0_(^P)(Γ(N)∩Γ_P(p);), then φ is cuspidal. When t^P_d> 2n+1 this is true according to <cit.>. 


As explained in <cit.>, for each pair (β_1,β_2)∈ N^-1(n,)^*⊕ 2_>0 there is a Λ_P-linear map

    ε_,β_1,β_2:^0_⊗__F T_P(_p)^0_Λ_P,

which, when specialized at primes _^P, gives the map of taking the coefficients indexed by β_1,β_2 in the p-adic q-expansion ε_.

Define

    μ__P,ϕ,β_1,β_2   = ε_,β_1,β_2∘1__P(μ_,)∈ eas(_p^×, F__P),
    ^imp__P,ϕ,β_1,β_2   =ε_,β_1,β_2∘1__P(^imp_)∈ F__P.



For a Dirichlet character ϕ with conductor dividing N and ϕ^2≠ 1, a geometrically irreducible component _P⊂Spec(^0,N_), and (β_1,β_2)∈ N^-1(n,)^*⊕ 2_>0, the above constructed μ__P,ϕ,β_1,β_2 and ^__P,ϕ,β_1,β_2 satisfy the following interpolation properties. Let x:__P F' be an F'-point of _P with F' being a finite extension of F. Suppose that the weight projection map Λ_P^n_d,N_ is étale at x mapping x to an arithmetic point ^P∈_(T_P(_p),F^'×).

If (κ,^P) is admissible, and the finite part of κ is χ, then

    (∫_^×_pκ dμ__P,ϕ,β_1,β_2)(x)=    C_x,β_1,β_2,ϕ,N· E_p(n+1-k,π_x×ϕχ) E_∞(n+1-k,π_x×ϕχ)
       × L^Np∞(n+1-k,π_x×ϕχ),

if κ(-1)=ϕ(-1) and x is classical, and otherwise vanishes.

Assume that the parity of _P is compatible with ϕ and that ^P is admissible. If x is classical, then

    ^imp__P,ϕ,β_1,β_2(x)=    C_x,β_1,β_2,ϕ,N· E^P-imp_p(n+1-t^P_d,π_x×ϕϵ^P_d) E_∞(n+1-t^P_d,π_x×ϕϵ^P_d)
       ×  L^Np∞(n+1-t^P_d,π_x×ϕϵ^P_d),

and if x is not classical, then we have 0=0.

The constant C_x,β_1,β_2,ϕ,N is defined as

    C_x,β_1,β_2,ϕ,N=    ϕ(-1)^nvol(Γ(N))∏_l=1^d∏_j=1^n_l(1-p^-j)/∏_j=1^n(1-p^-2j)
        √(-1)^n^2+n/22^n^2-n/22^-∑_j=1^d n_jt^P_x,j/((n),(^P_x)) ∑_φ∈_x(β_1,φ)(β_2,e_P(φ))/<φ,φ>,

where


  * _x is a finite set consisting of an orthonormal basis of the eigenspace for the Hecke eigensystem parametrized by x inside e_P  M^0_(^P)(Γ(N)∩Γ_SP(p^∞),;). The set _x is empty and the evaluation is 0 if x is not classical, i.e. if there exists no cuspidal irreducible automorphic representation π_x⊂_0(G()\ G()) with π_x,∞≅_(^P) such that the Hecke eigensystem parametrized by x appears in π_x.

  * (β_i,·), i=1,2, denotes the coefficient indexed by β_i in the q-expansion.

  * The operator  is defined as in (<ref>), and by the reasoning as in <cit.> we know that e_P(φ)≠ 0 for φ∈_x.



By our construction, for a classical x as in the theorem, the evaluations of 1__P(∫_^×_pκ dμ_,) and 1__P(^imp_) at x are classical cuspidal Siegel modular forms obtained by projecting the forms in (<ref>) to the eigenspace associated to the Hecke eigensystem parametrized by x. Thus the interpolation formulae follow from the formulae for

    _φ((e_P× e_P).E^*(·,f_κ,^P)|_G× G)   and      _φ((e_P× e_P).E^*(·,f_^P)|_G× G),

for φ∈_x, which are proved later in Proposition <ref>.


For each j∈(/(p-1) such that ϕω^j(-1)=1, applying the p-adic Mellin transform with respect to ω^j to the measure μ__P,ϕ,β_1,β_2, one gets the p-adic L-function __P,ϕω^j,β_1,β_2∈__P[[S]]⊗___PF__P as described in the theorem in the introduction.


Before finishing the calculations that prove the theorem, let us recall the state of the art concerning étaleness of eigenvarieties. First of all, the étale points are an open subset of Spec(_P). In general for a given point we don't have good criterion for étalness (except for n=1). The typical strategy to prove that a point x is étale over the weight space consists of two steps: first one shows that the system of eigenvalues corresponding to x correspond to a classical holomorphic Siegel modular forms, and then one uses automorphic methods to control the dimension of the space of eigenforms corresponding to the eigensystem x.

In our situation, for what concerns the first step, as we consider only admisible points the condition t_d^P ≥ n+1 ensure that the weights are cohomological and it is widely believed that all systems of eigenvalues of cohomological weights are classical, even though the best result on classicality at the moment <cit.> does not cover the case we study in the next section, where t_d^P =n+1, as they require t^P_d > n(n+1)/2.

For the second point, one needs to prove that δ(x)≤δ(y), where y is a point near x such that  the weight projection map is étale at y and where δ(x) (resp. δ(y)) is the dimension of the eigenspace corresponding to x (resp. y) inside ordinary Siegel modular forms of a fixed tame level. One approach for proving such an equality is to show that δ(x)=1. This can be done by adding _ℓ-operators for tame ramified places ℓ and assuming that they have distinct eigensystems at the point we consider, or for some cases of n=2 and Iwahori tame level, by using the exhaustive classification of local representations in <cit.>. For example, the dimension one result can be proved for the automorphic representation of GSp(4,) associated to the symmetric cube of π_f_X_0(11), for f_X_0(11) is the only normalized weight two elliptic cusp form of level Γ_0(11). Another possible approach for proving the inequality is to add suitable conditions at tame ramified places to guarantee that all eigenforms for x belongs to one irreducible automorphic representation, and then try to prove an analogue of <cit.> for symplectic groups.



 §.§ The archimedean zeta integral

Let E_∞(s,π×ξ) be the modified archimedean Euler factor for p-adic interpolation of critical values of L(s,π×ξ) to the left of the center according to the conjecture of Coates–Perrin-Riou. Unfolding the definition in <cit.>, if π_∞≅_ with =(t_1,…,t_n), then

    E_∞(s,π×ξ)= ∏_j=1^ne^-(s+t_j-j)π√(-1)/2Γ_(s+t_j-j)).



With f_κ,^P,∞∈ I_Q_H,∞(2n+1/2-k,sgn^k) as in (<ref>) and π_∞≅_(^P),

    Γ_2n(2n+1/2)/2^-2n^2+n+2nkπ^2n^2+n .Z_∞(f_κ,^P,∞(s),v^∨_(^P),v_(^P))|_s=2n+1/2-k
    
      =    √(-1)^n^2/2+n/22^n^2/2-n/2 2^-∑_j=1^d n_jt^P_j/((n),(^P)) E_∞(1+n-k,π×ϕχ).



According to <cit.>, the local zeta integral for π_∞ and f_κ,^P,∞(s,sgn^k)∈ I_Q_H,∞(s,sgn^k) satisfies the functional equation 

    Z_∞(f_κ,^P,∞(s),·,·)=Γ_∞(s,π_∞,sgn^k)^-1Z_∞(M(s,sgn^k)f_κ,^P,∞(s),·,·),

where M(s,sgn^k):I_Q_H,∞(s,sgn^k) I_Q_H,∞(-s,sgn^k) is the intertwining operator, and

    Γ_∞(s,π_∞,sgn^k)^-1
    
       =    π_∞(-1) γ_∞(s+1/2,π_∞×^k)^-1γ_∞(s-2n-1/2,^k) ∏_j=1^n γ_∞(2s-2j+2,).

It follows from <cit.> and the definition of f_κ,^P,∞ that

    M(s,sgn^k)f_κ,^P,∞(s,sgn^k)
    
       =    (-1)^nk 2^2n-2nsπ^n(2n+1)·Γ_2n(s)/Γ_2n(1/2(s+2n+1/2)+k/2)Γ_2n(1/2(s+2n+1/2)-k/2)  f_κ,^P∞(-s,sgn^k).

Combining (<ref>)(<ref>)(<ref>), we get

    .Z_∞(f_κ,^P,∞(s),·,·)|_s=2n+1/2-k=    √(-1)^k+n^2+n2^2nk-k-2n^2+n+1π^-(2n+1)k+2n^2+2nΓ_2n(k)Γ(k-n)/Γ_2n(2n+1/2)
       ×π_∞(-1)γ_∞(s+1/2,π_∞×^k) .Z_∞(f_κ,^P,∞(s),·,·)|_s=k-2n+1/2.

Let (t_1,…,t_n)=(^P). Easy computation shows that

    γ_∞(k-n,π_∞×^k)=√(-1)^-k+n^22^k-n-1π^k-n/Γ(k-n)π_∞(-1)∏_j=1^nΓ_(n+1-k+t_j-j)/Γ_(k-n+t_j-j),

and <cit.> says that

    .Z_∞(f_κ,^P,∞(s),v^∨_(^P),v_(^P))|_s=k-2n+1/2=√(-1)^nk-∑_j=1^nt_j2^-∑_j=1^n t_j+n^2+3n/2π^2nk/Γ_2n(k)((n),(^P)) ∏_j=1^nΓ_(k-n+t_j-j).

Plugging these two formulas into (<ref>) proves the proposition.






 §.§ Computing the zeta integrals at p

The goal of this section is to prove the following proposition, which will give the interpolation properties for our p-adic L-functions.


Let π⊂_0(G()\ G()) be an irreducible cuspidal automorphic representation with π_∞≅_(^P). Also, assume that π contains a P-ordinary Siegel modular form φ of weight (^P) and p-nebentypus ^P, invariant under the translation of (Γ(N)∩Γ_SP(p^∞))^∧⊂ G(_f). Then

    χ(-1)^nΓ_2n(2n+1/2)/2^2nk-2n^2+nπ^n+2n^2 _φ((e_P× e_P).E^*(·,f_κ,^P)|_G× G)
    
       =    ϕ(-1)^nvol(Γ(N))∏_l=1^d∏_j=1^n_l(1-p^-j)/∏_j=1^n(1-p^-2j) √(-1)^n^2-n/22^n^2-n/2 2^-∑_j=1^d n_jt^P_j/((n),(^P))
        × E_p(n+1-k,π×ϕχ) E_∞(n+1-k,π×ϕχ)  L^Np∞(n+1-k,π×ϕχ)· e_P(φ),

and 

    ϵ^P_d(-1)^nΓ_2n(2n+1/2)/2^2nt^P_d-2n^2+nπ^n+2n^2 _φ((e_P× e_P).E^*(·,f_^P)|_G× G)
    
       =    ϕ(-1)^nvol(Γ(N))∏_l=1^d∏_j=1^n_l(1-p^-j)/∏_j=1^n(1-p^-2j) √(-1)^n^2-n/22^n^2-n/22^-∑_j=1^d n_jt^P_j/((n),(^P))
        × E^P-imp_p(n+1-t^P_d,π×ϕϵ^P_d) E_∞(n+1-t^P_d,π×ϕϵ^P_d)  L^Np∞(n+1-t^P_d,π×ϕϵ^P_d)· e_P(φ).



With the results in the last section, the proof of the proposition is mainly about computing the zeta integrals at p. Because of the ordinary projection e_P× e_P, we need to compute at p the zeta integrals for R(U^P_p)× R(U^P_p)f_κ,^P and R(U^P_p)× R(U^P_p)f_^P, where R is a polynomial depending on π_p and a sufficiently small open compact subgroup K_p⊂ G(_p) satisfying e_P=R(U^P_p) on π^K_p. Recall that with a fixed ^P, we make ^P_p-operators act on smooth G(_p)-representations by (<ref>).

The computation will essentially use Proposition <ref> and is similar as the computation in <cit.>.


Since the projections to π_p of P-ordinary forms inside π span a one-dimensional subspace, we know that _φ((e_P× e_P).E^*(·,f_κ,^P)|_G× G) (resp. _φ((e_P× e_P).E^*(·,f_^P)|_G× G)) equals e_P(φ) up to a scalar, given by 

    <_φ((e_P× e_P).E^*(·,f_κ,^P)|_G× G),φ^'>/<e_P(φ),φ^'>,    (resp.  <_φ((e_P× e_P).E^*(·,f_^P)|_G× G),φ^'>/<e_P(φ),φ^'>)

for an arbitrary φ'∈π such that the denominator is nonzero. We will always assume that φ' is fixed by both K_p and SP_G(_p) and e_Pφ'≠ 0. Then

    <e_P(φ),φ^'>=<φ,e_Pφ'>≠ 0.

Let =(b_1,…,b_1_n_1,…,b_d,…,b_d_n_d) for b_1>b_2>⋯>b_d>0 and denote diag(p^b_1,…,p^b_1,…,p^b_d,…,p^b_d) by p^. A direct computation shows that

    <_φ((e_P× U^P_p,).E^*(·,f_κ,^P)|_G× G),φ^'>
    
       =    ϕ(-1)^nvol(Γ(N))·Z_∞(f_κ,^P,∞,v_(^P),v^∨_(^P))/<v_(^P),v^∨_(^P)>· L^Np∞(n+1-k,π×ϕχ)
        .× p^<+2ρ_G,c, ><φ,R(U^P_p) ∫_G(_p)f^α_κ,^P_p(s,ϕχ)(^-1ι(g^-1,1))π_p(gp^)φ' dg>|_s=2n+1/2-k.

The ratio

    <φ,R(U^P_p) ∫_G(_p)f^α_κ,^P_p(s,ϕχ)(^-1ι(g^-1,1))π_p(gp^)φ' dg>/<φ,e_Pφ'>

is independent of the choice of φ. Let θ_1,…,θ_n, α_1,…,α_n be as in Proposition <ref>, and σ_1,…,σ_d be as in the definition (<ref>). We know that the P-ordinary space (with respect to ^P) inside Ind^G(_p)_B_G(_p) is one dimensional. Therefore, if we take from Ind^G(_p)_B_G(_p) a function :G(_p) invariant under the right translation by K_p and SP_G(_p), then

    (<ref>)=(R(U^P_p) ∫_G(_p)f^α_κ,^P_p(s,ϕχ)(^-1ι(g^-1,1))π_p(gp^) dg)(1)/(R(U^P_p))(1),

as long as the denominator is nonzero. Now we further assume that (1)=1. Then by our description of the ^P_p-eigenvalues for the P-ordinary vector in terms of the α_i's in Proposition <ref>, we have (R(U^P_p))(1)=1. 

Now let ∈Ind^G(_p)_B_G(_p) be the smooth function on G(_p) supported on B_G(_p) SP_G(_p) and taking the value 1 on SP_G(_p). We also put (a)=| a|^-n+1/2_p([    a    0;    0 a^-1 ]). Then ∈Ind^(n,_p)_B(_p) and is invariant under the right translation of u for  u∈ SP(_p) and takes the value 1 at identity. Let Φ_χ,^P be the Schwartz function on M_n(_p) whose Fourier transform is  

    Φ_χ,^P(β_0)   =∏_i=1^d1_(N_i,_p)((β_0)_N_i)× ∏_i=1^d-1ϵ^P_iϵ^P-1_i+1(det(-(β_0)_N_i))·ϵ^P_dχ^-1(det(-β_0)),   β_0∈ M_n(_p).

An easy computation shows that 

    f^α_κ,^P_p(s,ϕ_pχ_p)(^-1ι(a   b
    c   d,1))=   | a|_p^-(s+2n+1/2)ϕχ((-a))^-1p^-n(n+1)
        ×1_(n,_p)(a)·1_p^-2(n,_p)(a^-1b)1_(n,_p)(ca^-1)·Φ_χ,^P(a^-1),

and

    (U^P_p,∫_G(_p)f^α_κ,^P_p(s,ϕ_pχ_p)(^-1ι(g^-1,1))π_p(gp^) dg)(1)
    
       =    ∏_j=1^n(1-p^-j)/∏_j=1^n(1-p^-2j)p^-<ρ_G,n, >∏_i=1^d a^c_i_N_i∫_(n,_p)| a|_p^s+n/2ϕχ((-a))Φ_χ,^P(a)(ap^) da,

for =(c_1,…,c_1_n_1,…,c_d,…,c_d_n_d), c_1>c_2>⋯>c_d>0.

Therefore, in order to show (<ref>), it suffices to show that with our chosen ∈Ind^(n,_p)_B(_p), 

    ∫_(n,_p)| a|_p^s+n/2   ϕ_pχ_p( a)Φ_χ,^P(a)(ap^) da
       =∏_i=1^d∏_j=1^n_i(1-p^-j)/∏_j=1^n(1-p^-j)∏_i=1^d ∏_j=N_i-1+1^N_iα_j^b_i· E_p(s,π×ϕχ).

By the same reasoning, in order to show (<ref>), it suffices to show 

    ∫_(n,_p)| a|_p^s+n/2   ϕ_pϵ^P_d,p( a)Φ_^P(a)(ap^) da
       =∏_i=1^d∏_j=1^n_l(1-p^-j)/∏_j=1^n(1-p^-j)∏_i=1^d ∏_j=N_i-1+1^N_iα_j^b_i· E^P-imp_p(s,π×ϕϵ^P_d),

where Φ_^P is the Schwartz function on M_n(_p) whose Fourier transform is

    Φ_^P(β_0)   =1_M_n(_p)(β_0)∏_i=1^d-11_(N_i,_p)((2β_0)_N_i)× ∏_i=1^d-1ϵ^P_iϵ^P-1_i+1(det(-(2β_0)_N_i)),   β_0∈ M_n(_p).


We will show (<ref>) and (<ref>) by induction. Write n'=N_d-1, a=ccc   n'   n_d
c(cc)n'   a'   η
n_d   0   λ
·[  I_n'     0; λ^-1μ I_n_d ]. Define '∈Ind_B_n'(_p)^(n',_p)(θ_1,…,θ_n') by '(a')=| a'|_p^-n_d/2([    a'     0;     0 I_n_d ]), and the (n_d,_p)-fixed _d∈Ind^(n_d,_p)_B_n_d(_p)(θ_n'+1,…,θ_n) by _d(λ)=| a|_p^n'/2([ I_n'    0;    0    λ ]). Also, let

    Φ'_χ,^P(β'_0)   =∏_i=1^d-11_(N_i,_p)((β'_0)_N_i)× ∏_i=1^d-2ϵ^P_iϵ^P-1_i+1(det(-(β'_0)_N_i))·ϵ^P_d-1χ^-1(det(-β'_0)),   β'_0∈ M_n'(_p),
    Φ'_^P(β'_0)   =∏_i=1^d-11_(N_i,_p)((β'_0)_N_i)× ∏_i=1^d-2ϵ^P_iϵ^P-1_i+1(det(-(β'_0)_N_i))·ϵ^P_d-1ϵ^P-1_d(det(-β'_0)),   β'_0∈ M_n'(_p),

and let Φ'_χ,^P, Φ'_^P be the inverse Fourier transform of Φ'_χ,^P, Φ'_^P. Denote by _^P_dχ^-1 the inverse Fourier transform of the Schwartz function λ↦1_(n_d,_p)(λ)·^P_dχ^-1((-λ)) on M_n_d(_p). Then by a routine computation we get

    LHS of (<ref>)=   ∏_j=1^n'(1-p^-j)∏_j=1^n_d(1-p^-j)/∏_j=1^n(1-p^-j)∫_(n',_p)| a'|_p^s+n'/2ϕ_pχ_p( a')'(a'p^^')Φ'_χ,^P(a') da'
        ×∏_i=N_d-1+1^nα_i^b_d∫_(n_d,_p)|λ|_p^s+n_d/2ϕ_pχ_p(λ) _d(λ)_ϵ^P_dχ^-1(λ) dλ,
    LHS of (<ref>)=   ∏_j=1^n'(1-p^-j)∏_j=1^n_d(1-p^-j)/∏_j=1^n(1-p^-j)∫_(n',_p)| a'|_p^s+n'/2ϕ_pχ_p( a')'(a'p^^')Φ'_^P(a') da'
        ×∏_i=N_d-1+1^nα_i^b_d∫_(n_d,_p)|λ|_p^s+n_d/2ϕ_pχ_p(λ) _d(λ) 1_M_n_d(_p)(λ) dλ.


For _ϵ^P_dχ^-1, we have the following formula <cit.>. If cond(ϵ^P_dχ^-1)=p^c>1, 

    _ϵ^P_dχ^-1(λ)   =p^-n_d(n_d+1)c/2G(ϵ^P_dχ^-1)^n_d·1_(n_d,_p)(p^cλ)ϵ^P-1_dχ((p^cλ)),

and if ϵ^P_d=χ,

    _triv(λ)=∑_j=0^n_d(-1)^jp^j(j-1-n_d)/2∫_(n_d,_p)pI_j   0
    0   I_n_d-j(n_d,_p)1_M_n_d(_p)(λ g) dg.

We first look at the easier case (<ref>), where for the integral in the second line of it, since _d is fixed by (n_d,_p) and (n_d,_p) acts on it by ϵ^P-1_d,p∘, we have

    ∫_(n_d,_p)|λ|_p^s+n_d/2ϕ_pϵ^P_d,p(λ) _d(λ) 1_M_n_d(_p)(λ) dλ=L_p(s+1/2,σ_d⊗ϕ_p).

Next we treat the integral in the second line of (<ref>) with ϵ^Pχ^-1≠triv. Plugging in (<ref>), we get

    ∫_(n_d,_p)|λ|_p^s+n_d/2χ_p(λ) _d(λ)_ϵ^P_dχ^-1(λ) dλ=∏_j=n'+1^n G(ϵ^P_dχ^-1)(ϕ(p)^-1α^-1_jp^s-1/2)^c
    
       =    ∏_j=n'+1^nγ_p(1/2-s,ϕ^-1_pχ^-1_pθ^-1_j)=γ_p(1/2-s,σ_d⊗ϕ^-1_pχ^-1_pϵ^P_d,p).

Lastly, we consider the integral in the second line of (<ref>) with ϵ^Pχ^-1= triv. The formula for the Hecke action on spherical representations of (_p) gives

    ∫_(n_d,_p)p^-1I_j   0
    0   I_n_d-j(n_d,_p)_d(λ g) dg=p^j(n_d-j)/2∑_ι:{1,…,j}{1,…,n_d}
    ι(1)<…<ι(j)α^-1_n'+ι(1)…α^-1_n'+ι(j)_d(λ).

Combining this with (<ref>), we get

    ∫_(n_d,_p)|λ|_p^s+n_d/2χ_p(λ) _d(λ)_triv(λ) dλ
    
      =    ∏_j=n'+1^n(1-ϕ_p(p)^-1α^-1_jp^s-1/2)∫_(n_d,_p)|λ|_p^s+n_d/2ϕ_pϵ^P_d,p(λ) _d(λ) 1_M_n_d(_p)(λ) dλ
    
      =   ∏_j=n'+1^n(1-ϕ_p(p)^-1α^-1_jp^s-1/2)· L_p(s+1/2,σ⊗ϕ_p)=γ_p(1/2-s,σ_d⊗ϕ^-1_p).

The identities (<ref>), (<ref>), and (<ref>) allows us to compute (<ref>) and (<ref>) by induction, and this finishes the proof.




§ THE DERIVATIVE OF THE P-ADIC STANDARD L-FUNCTION


In this section we explicit the trivial zeros of the p-adic L-function μ__P,ϕ,β_1,β_2 of Theorem <ref> and interpret them from the point of view of the (conjectural) associated p-adic Galois representation. This will allow us to interpret a factor appearing in the derivative of the p-adic L-function as Greenberg's ℓ-invariant in the case when the trivial zero is semi-stable (or of type M as called in <cit.>). At the end, we shall prove the main theorem of the paper which relates the derivative of __P,ϕω^n+1,β_1,β_2 at the semi-stable trivial zero to the ℓ-invariant and the complex special value.



  



 §.§ Greenberg-Benois conjecture and l-invariants



That idea at the base of the conjecture by Greenberg and Benois is that when a p-adic L-function has trivial zeros one should be able to recover the value of the complex L-function from a suitable derivative of the p-adic L-function. The first studied case is the one of an elliptic curve with split multiplicative reduction at p <cit.>. They conjectured that the order of the p-adic L-function is 1 plus the order of the complex L-function and that the leading coefficient of the p-adic L-function is the same as the one of the complex L-function, up to an error term equal to log_p(q_E)/ord_p(q_E) (for q_E the Tate uniformizer of E) that  they call the ℓ-invariant. 

Later Greenberg interpreted this number in Galois theoretical terms and proposed a similar conjecture for a great number of ordinary motives  <cit.>; succesively Benois generalised this conjecture to most semistable representations <cit.>. Let us be more precise. Let V the p-adic Galois representation associated with the motive, and suppose that it is absolutely irreducible and satisfies Pantchishkine condition, i.e. that there is a sub space V' of V stable under the action of G_ Q_p and containing all positive Hodge–Tate weights. Assume moreover that the Frobenius acts semisimply on the semi-stable module (in the sense of Fontaine) associated with V. 

Greenberg defines two subspaces V^11⊂ V^+ ⊂ V^00 such that V^+/V^11 contains all the eigenvalues p and V^00/V^+ contains all the eigenvalues 1. Then we can decompose V^00/V^11 as Q_p^t_0⊕ M ⊕ Q_p(1)^t_1, where M is a non split extension of Q_p^t by Q_p(1)^t. According to Greenberg's conjecture, the number of trivial zeros of the p-adic L-function is e=t_0+t+t_1. Assume furthermore that t_1=0 (certain motivic conjectures imply t_1t_0=0, so this hypothesis is not really restrictive as if t_0=0 can always consider the dual motive for which t_0=0). Then by picking a subspace 𝐓⊂ H^1(G_,V) of dimension e, Greenberg defines an e-dimensional subspace 𝐓_p inside H^1(G_ Q_p,V)/H^1_ f(G_ Q_p,V) which injects into 
    H^1(G_ Q_p,V^1/V')≅⊕^t+e_0_i=1 Q_p ·ord_p ⊕ Q_p ·log_p.
  
If we denote by p_u (resp. p_c) the projection of 𝐓_p to  ⊕^t+e_0_i=1 Q_p ·ord_p (resp. ⊕^t+e_0_i=1 Q_p ·log_p), then Greenberg shows that p_c is an invertible map and defines ℓ(V):=det(p_u ∘ p_c^-1). Note that in general 𝐓_p depends on V as G_ Q-representation, but if e_0=0 too then it depends only on the restriction to G_ Q_p <cit.>, <cit.>. We can finally state:
[Greenberg–Benois conjecture]
Let r be  the order of the complex L-function L(s,V) at s=0, and suppose that _p(S,V) has e_0+t trivial zeros, then S^e_0+t+r divides _p(S,V) exactly and

    _p(S,V)/S^e_0+t+r≡ℓ(V^*(1)) (E_p(0,V)L_p(0,V)^-1)^* L^alg,(r)(0,V)/log_p(1+p)^e_0+t+r (e_0+t+r)! S,

where E_p(s,V) is defined as in <cit.>, V^*(1) is the dual representation twisted by 1, L_p(s,V) is the Euler factor of the motivic L-function for V and (E_p(s,V)L_p(s,V)^-1)^* is obtained from E_p(s,V)L_p(s,V)^-1 by eliminating all the Euler factors vanishing at s=0. The function L^alg(s,V) is the L-function for V divided by the period.

Note that the conjecture implies the non-vanishing of the ℓ-invariant ℓ(V).

This conjecture has been first shown in the case of an elliptic curve with bad multiplicative reduction by Greenberg–Stevens <cit.>. Their method has been generalised many times to different contexts <cit.> and is at the base of our current approach. It is worth to point out that many new strategies have recently arisen <cit.>.



 §.§ The trivial zero of the p-adic standard L-function

Let __P,ϕω^j,β_1,β_2=__P,ϕ,β_1,β_2(S,x)∈__P[[S]]⊗___PF__P to be the Mellin transform of the component corresponding to the character ω^j on (/p)^× of the measure μ__P,ϕ,β_1,β_2 constructed in Theorem <ref>.

Suppose that x_0:__P F is a point as in Theorem <ref> and it is classical. Denote by π_x_0 a cuspidal irreducible automorphic representation of (2n,) generated by a P-ordinary Siegel modular ^0,N_-eigenform of weight ι(^P) whose eigenvalues are parametrized by x_0. The isomorphism class of π_x_0,v is determined by x_0 for v∤ N (the isomorphism class of π_x_0,p can be read off from the eigenvalues for all ^P_p-operators by the discussion in <ref>). We are interested in possible trivial zeros of __P,ϕω^n+1,β_1,β_2 at S=(1+p)^n+1-1, where the corresponding critical L-value is the near-central value L(0,π_x×ϕ). There are two types of trivial zeroes that can show up there.



  §.§.§ Crystalline trivial zero
 
We say that __P,ϕω^n+1,β_1,β_2 has a crystalline trivial zero at ((1+p)^n+1-1,x_0) if  ϕ_p(p)=1 and the local L-factor L_p(s,π_x_0×ϕ_p) contains the factor (1-ϕ_p(p)p^-s). In particular, if π_x_0,p is unramified, then a crystalline trivial zero shows up at S=(1+p)^n+1-1. 

One can also think of types of trivial zeroes in terms of the corresponding Galois representation restricted to G__p (if admitting the local-global compatibility for ρ_x_0 attached to π_x_0). Let ρ_x_0:G_(2n+1,_p) be the Galois representation attached to π_x_0 <cit.>. Then conjecturally ρ_x_0|_G__p admits the following description. The Hodge–Tate weights are 0,± (t^P_1-1),…,± (t^P_1-N_1),±(t^P_2-N_1-1),…,±(t_2-N_2),…,± (t^P_d-N_d-1-1),…,± (t^P_d-n), and the eigenvalues of the Frobenius are 1,α^± 1_1,…,α^± 1_n. Hence the Hodge polygon and the Newton polygon meet at the points with horizontal coordinates 0, N_1, N_2,…,N_d,2n+1-N_d,…,2n+1-N_2,2n+1-N_1,2n+1. Therefore, ρ_x_0 admits a decreasing filtration Fil^j, -d≤ j≤ d, such that Fil^j/Fil^j+1 has Hodge–Tate weights t^P_d-j+1-N_d-j-1,…,t^P_d-j+1-N_d-j+1 (resp. t^P_-j-N_-j+1-1,…,t^P_-j-N_-j) if 1≤ j≤ d (resp. -d≤ j≤ -1), and Fil^0/Fil^1 is one dimensional with trivial G__p-action. If Fil^0/Fil^2 is a trivial extension of _p by Fil^1/Fil^2, then __P,ϕω^n+1,β_1,β_2 has a crystalline trivial zero at ((1+p)^n+1-1,x_0) if  ϕ_p(p)=1.



  §.§.§ Semi-stable trivial zero

The other case is when the local L-factor L_p(s,π_x_0×ϕ_p) does not contain the factor (1-ϕ_p(p)p^-s), but the factor (1-ϕ_p(p)^-1α_n^-1 p^s-1) in E_p(s,π_x_0×ϕ) contributes a trivial zero at ((1+p)^n+1-1,x_0) when ϕ_p(p)=1 and α_n=p^-1. We call this type of trivial zero semi-stable. Since Newton polygon lies above the Hodge polygon and for ρ_x_0|_G__p they coincide along the slope 0 segment, we have v_p(α_n)≤ -t^P_d+n≤ -1. If α_n=p^-1, then t^P_d=n+1 and the Newton and Hodge polygons coincide along the the segments of slope -1,0,1. Therefore, if __P,ϕω^n+1,β_1,β_2(S,x_0)∈_F[[S]]⊗__F F has a semi-stable trivial zero at S=(1+p)^n+1-1, then there exists a partition n=n_1+… +n_d with n_d=1 such that π_x_0 is also P'-ordinary for P'⊂(n) the standard parabolic subgroup attached to this partition. 

The above discussion shows that in order to use p-adic deformation of π_x_0 to study the trivial zero at S=(1+p)^n+1-1 of the p-adic L-function attached to π_x_0, one can always choose P such that n_d=1 and consider the P-ordinary families passing through x_0. In the following, we assume that n_d=1. 

In terms of the local Galois representation, a semi-stable trivial zero appears if Fil^0/Fil^2 is a non-trivial extension of _p by _p(1).



 §.§ A formula for the derivative

In this section we prove the main theorem of the paper applying the strategy of Greenberg–Stevens. Recall that a_i, 1≤ i ≤ n, denotes the eigenvalue of the operator U^P_p,i (see Proposition <ref>). They are invertible elements inside __P. In order to state <cit.> in our setting, we need to fix a coordinate for a rigid analytic open neighborhood of x_0 in _P. Since we have assumed that the weight projection map is étale at x_0, we can take the coordinates to be the natural coordinates T^P_1,…,T^P_d of the weight space. Then it follows from <cit.> and <cit.> that 

Let x_0∈_P(F) be a classical point where the weight projection map is étale and has image ^P_x_0∈_(T^P(_p),F^×). Suppose that S=(1+p)^n+1-1 is a semi-stable trivial zero for __P,ϕω^n+1,β_1,β_2(S,x_0) and the local-global compatibility is satisfied by the p-adic Galois representation ρ_x_0. Then 

    ℓ(ρ_x_0)=ℓ(ρ_x_0^*(1)) =- . ∂log_p (a_n(T^P_1,… T^P_d)/a_n-1(T^P_1,…,T^P_d))/∂ T^P_d|_(T^P_1,…,T^P_d)=^P_x_0(1+p).


(For the proof see  <cit.>, where the theorem is stated for parallel weight, but the proof is the same.)


When n=2 one can calculate the ℓ-invariant also for crystalline trivial zero using the method of <cit.>.


The p-adic L-function __P,ϕω^n+1,β_1,β_2 vanishes identically on the hyper-plane S=(1+p)^n+1-1 whenever ϕ_p(p)=1. 


We shall show that if ϕ_p(p)=1 then μ_, vanishes at all interpolation points with S=(1+p)^n+1-1. This will imply that μ_, vanishes on the hyper-plane S=(1+p)^n+1-1, so __P,ϕω^n+1,β_1,β_2 will also vanish on that hyper-plane.

Indeed, from Proposition <ref> we see that all Fourier coefficients _κ,^P() present the factor L^Np∞(0,ϕλ_); we recall that λ_(p) is the Legendre symbol ((-1)^rank()/2^*()/p), where ^*() denotes the product of all the nonzero eigenvalues of . 

Note that in the construction of μ_,, our choice of local sections at p makes W_,p vanish unless =[ β_1 β_0; β_0 β_2 ] with β_0∈(n,_p) and β_1≡ 0 p (see Table <ref>). 

Such 's are of maximal rank 2n, and hence 

    ^*()=()≡[   0 β_0; β_0 β_2 ]≡ (-1)^n(β_0)^2  p.
  Thus, (-1)^rank()/2()
 belongs to ^×_p and 
 
    (-1)^rank()/2() ≡(-1)^rank()/2 (-1)^n(β_0)^2  ≡(β_0)^2  p.


It follows that λ_(p)=1. Therefore, as we also have ϕ_p(p)=1, we get

    L^Np∞(0,ϕλ_) = (1-λ_(p)ϕ_p(p)p^0)L^N∞(0,ϕλ_)=0,

which implies the vanishing of all the _κ,^P()'s and the vanishing of the measure μ_, along S=(1+p)^n+1-1.

 It is possible to define an algebraic ℓ-invariant ℓ(_P), which is a meromorphic function on _P, using the definition of <cit.> in the context of <cit.>.

Define j(_P)∈/(p-1) by ω^j(_P)=^P_d|_(/p)^× for a ^P inside the image of the projection of _P to the weight space. The relation of the improved p-adic L-function ^__P,ϕ,β_1,β_2 and the restriction of __P,ϕω^j(_P),β_1,β_2 to κ=^P_d is given as:

Suppose that n_d=1. As elements in F__P we have 

    . __P,ϕω^j(_P),β_1,β_2|_κ=^P_d=(1-ϕ_p(p)^-1a_n-1/a_n )^__P,ϕ,β_1,β_2.



This follows straightforwardly from the two interpolation formulae of Theorem <ref> by noticing that when the point x is classical and n_d=1, ^P(π_x×ξ)=(1-ϕ_p(p)^-1α_n^-1p^n-t^P_d), so

    E_p(n+1-t^P_d,x,π_x×ϕϵ^P_d)= (1-ϕ_p(p)^-1a_n-1(x)/a_n(x) )E^P-imp_p(n+1-t^P_d,x,π_x×ϕϵ^P_d)

(cf. (<ref>)).


Now we are ready to prove the main theorem. 

Let x_0 be an F-point of _P where the weight projection map Λ_P^1,N_ is étale and maps x_0 to ^P_0. Suppose that x_0 is classical and that the p-adic L-function __P,ϕω^n+1,β_1,β_2∈__P[[S]] ⊗___P F__P has a semi-stable trivial zero at ((1+p)^n+1-1,x_0) (so j(_P)=n+1) and the local-global compatibility is satisfied by the p-adic Galois representation ρ_x_0. (This in particular implies that ϵ^P_d is trivial as ϕ has level prime to p.) Then we have 

    .__P,ϕω^n+1,β_1,β_2(S,x_0)/ S|_S=(1+p)^n+1-1
    
       =     -ℓ(ρ_x_0)· C_x_0,β_1,β_2,ϕ,N· E^P-imp_p(0,π_x_0×ϕ)  E_∞(0,π_x_0×ϕ)  L^Np∞(0,π_x_0×ϕ),



Again, we use the natural coordinate T^P_1,…, T^P_d of the weight space to parametrize a rigid analytic open neighborhood of x_0 in _P. Note that by Lemma <ref>, we know that __P,ϕω^n+1,β_1,β_2((1+p)^n+1-1,x) is identically vanishing, so  

    .∂__P,ϕω^n+1,β_1,β_2(S,T^P_1,…,T^P_d)/∂ T^P_d|_S=(1+p)^n+1-1=0.

It follows that

    .__P,ϕ,β_1,β_2(S,x_0)/ S|_S=(1+p)^n+1-1
    
       =   .(∂__P,ϕω^n+1,β_1,β_2(S,T^P_1,…,T^P_d)/∂ S+∂__P,ϕω^n+1,β_1,β_2(S,T^P_1,…,T^P_d)/∂ T^P_d)|_S=(1+p)^n+1-1,(T^P_1,…,T^P_d)=^P_0(1+p)
    
       =   .∂/∂ T^P_d((1-ϕ_p(p)^-1a_n-1(T^P_1,…,T^P_d)/a_n(T^P_1,…,T^P_d) )^imp__P,ϕ,β_1,β_2(T^P_1,…,T^P_d))|_(T^P_1,…,T^P_d)=^P_0(1+p)
    
       =   . ∂(a_n(T^P_1,… T^P_d)/a_n-1(T^P_1,…,T^P_d))/∂ T^P_d|_(T^P_1,…,T^P_d)=^P_0(1+p)· ^imp__P,ϕ,β_1,β_2(x_0).

Then the theorem follows from Theorem <ref> and the interpolation property of ^imp__P,ϕ,β_1,β_2 in Theorem <ref>.




alpha ]