Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula
Samuele Anni Gebhard Boeckle Peter Mathias Graef Alvaro Troya



Transiting Exoplanet Monitoring Project (TEMP). I. Refined System Parameters and Transit Timing Variations of HAT-P-29b
    Gregory Laughlin
    
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In this article, we describe how to compute slopes of p-adic Ł-invariants of arbitrary weight and level by means of the Greenberg-Stevens formula. Our method is based on the work of Lauder and Vonk on computing the reverse characteristic series of the U_p-operator on overconvergent modular forms. Using higher derivatives of this characteristic series, we construct a polynomial whose zeros are precisely the Ł-invariants appearing in the corresponding space of modular forms with fixed sign of the Atkin-Lehner involution at p. In addition, we describe how to compute this polynomial efficiently. In the final section, we give computational evidence for relations between slopes of Ł-invariants for small primes.




§ INTRODUCTION


Let f be a newform of even weight k≥ 2 for Γ_0(pN), where p is prime and (N,p)=1. Let χ be a Dirichlet character of conductor prime to pN with χ(p)= 1. By the work of Mazur and Swinnerton-Dyer, see for example <cit.>, there exists a p-adic L-function L_p(f,χ,s) attached to f that interpolates the algebraic parts L^alg(f,χ,j) for j∈{1,…,k-1} of the special values of the classical L-function attached to f. The p-adic ℒ-invariant ℒ_p(f)∈_p attached to f satisfies 

    L'_p(f,χ,k2)=ℒ_p(f)· L^alg(f,χ,k2)

and it depends only on the local Galois representation attached to f. Mazur, Tate and Teitelbaum conjectured that such invariant should exist. Afterwards, several possible candidates were proposed. The fundamental breakthrough in relating these candidates to one another and proving the above formula, due to Greenberg and Stevens <cit.>, is the connection between ℒ_p(f) and the (essentially unique) p-adic family of eigenforms passing through f. If we denote this family by 

    f_κ=∑_n=1^∞ a_n(κ)q^n   with   f_k=f,

where the coefficients a_n(κ) are rigid analytic functions on a disc containing k in the weight space Hom_cont(_p^×,𝔾_m), then the relation is given as

    ℒ_p(f)=-2dlog(a_p(κ))|_κ=k.

A more detailed account of the history of ℒ-invariants can be found in <cit.>.

Recently, there has been a lot of interest in computing p-adic ℒ-invariants. In <cit.>, the third author conjectured some relations between (slopes of) ℒ-invariants of different levels and weights for p=2. In <cit.>, Bergdall explains relations between slopes of ℒ-invariants and the size of the p-adic family passing through the given newform.
On a different note, a main motivation to understand and compute ℒ-invariants arises from the study of coefficient fields of classical newforms. In particular, Buzzard asked whether there exists a bound B_N,p for all Hecke eigenforms f of level Γ_1(N)∩Γ_0(p) and any weight k≥ 2 such that [_f,p : _p]≤ B_N,p, where _f,p is the coefficient field of f completed at a prime dividing p, see <cit.>. 
For N = 1 and p ≤ 7 Chenevier has shown (<cit.>) that _f,p is either 
_p(a_p(f)) if f is old at p, or _p (ℒ_p(f)) if f is new at p, where a_p(f) is the p-th coefficient of the q-expansion of f at infinity. Therefore, Buzzard's bound B_N,p would constrain the degrees of ℒ_p(f) and a_p(f) over _p for all f as long as N and p are fixed. 

In this article, we describe a procedure for computing ℒ-invariants via the Greenberg-Stevens formula, building on the work of Lauder and Vonk on  the action of the U_p-operator on  overconvergent modular forms. In <cit.> Lauder describes how to compute the ℒ-invariant from the first derivative of the (inverse)  characteristic series of the U_p-operator if there is a unique split multiplicative cusp form in the given weight, using a formula of Coleman, Stevens and Teitelbaum <cit.>. The main aim of this article is to extend this method to weights k in which there is more than one split multiplicative cusp form. For this purpose, we explain how to efficiently compute higher derivatives of the characteristic series and how the ℒ-invariants can be read off from these derivatives. In order for this approach to work and not increase too much the precision required to perform the computations, one first needs to decompose the space of p-newforms by their Atkin-Lehner eigenvalue at p. Since the needed precision grows with the dimension of these subspaces, we give explicit dimension formulas, analogous to the ones presented in <cit.>.Let us remark that the procedure presented in <cit.> computes more data than the present paper, as  the ℒ-invariants in <cit.> are realized as the eigenvalues of  a "Hecke operator", called the ℒ-operator, whose matrix is computed explicitly. However, the procedure is naturally more restricted, since it involves the Jacquet­-Langlands correspondence and, therefore, it needs an auxiliary prime in the level. Another approach (unpublished) due to R. Pollack is to compute the ℒ-invariants directly in terms of p-adic and classical L-values via modular symbols.  Yet another approach is due to X. Guitart and M. Masdeu, see <cit.>.

The layout is as follows. In Section <ref> we recall briefly Coleman classicality and the main result of <cit.>, which are going to be used in several central parts of this article. In Section <ref>, we describe how to generalize a formula of Coleman, Stevens and Teitelbaum <cit.> proving the existence of a polynomial 𝒬_p,k, built from higher derivatives of the reverse characteristic series of the U_p operator on the space of overconvergent p-adic cusp forms of level Γ_0(pN), and whose roots are precisely the ℒ-invariants of level Γ_0(pN). In Sections <ref> and <ref> we show how to compute this polynomial extending a method of Lauder <cit.> and we give dimension formulae for the relevant spaces of classical modular forms.
In Section <ref> we present the data collected together with observations on the slopes.

All computations have been done using a  implementation of the algorithms. The code relies on algorithms of Lauder and Vonk and is available on request.

[Acknowledgements]
This work was partially supported by the DFG Forschergruppe 1920, the DFG Priority Program SPP 1489 and the Luxembourg FNR. 
We would also thank Alan Lauder, Robert Pollack and Jan Vonk for helpful remarks and suggestions. Finally, we would like to thank John Cremona for providing access to the servers of the Number Theory Group at the Warwick Mathematics Institute.




§ CLASSICAL AND OVERCONVERGENT MODULAR FORMS


Throughout this article, let p be prime and let N be a positive integer coprime to p. Let 𝒲 denote the even p-adic weight space. Thus, the _p-points of 𝒲 are continuous characters κ_p^×→_p^× with κ(-1)=1. For each κ∈𝒲, let 𝒮^†_κ(Γ_0(pN)) denote the space of overconvergent p-adic cusp forms of weight κ, see <cit.>. This space is equipped with the action of the compact operator U_p. 
For each integer k we realize the space of classical cuspforms 𝒮_k(Γ_0(pN)) as a U_p-stable subspace of 𝒮^†_k(Γ_0(pN)) by considering the associated weight z↦ z^k in 𝒲. Let

    P(k,t):=(1-t  U_p|𝒮_k(Γ_0(pN))),
       P^†(κ,t):=(1-t U_p|𝒮^†_κ(Γ_0(pN))),

be the reverse characteristic polynomial (respectively series) of U_p. We denote by 𝒪(𝒲){{ t}} the ring of entire functions with coefficients in 𝒪(𝒲). Then, if we write 

    P^†(κ,t)=1+∑_n=1^∞ b_n(κ)t^n∈𝒪(𝒲){{ t}},

by <cit.> each function κ↦ b_n(κ) is defined by a power series with coefficients in _p. More precisely, if we write 𝒲=_ϵ𝒲_ϵ, where the union is formed over the even characters ϵ (/2p)^×→_p^×, then we have κ∈𝒲_ϵ if and only if the restriction of κ to the torsion subgroup of _p^× is ϵ. For a fixed topological generator γ of 1+2p_p, each 𝒲_ϵ is an open p-adic unit disc with coordinate w_κ=κ(γ)-1. Then, for each such ϵ, there exists 

    P^†,ϵ(w,t)=1+∑_n=1^∞ b_n^ϵ (w)t^n∈_p[[w]]{{t}}

such that P^†(κ,t)=P^†,ϵ(w_κ,t) for κ∈𝒲_ϵ. In the sequel, we surpress ϵ in the notation. The following theorem, due to Coleman, links P(k,t) and P^†(k,t).  For a detailed account in the cuspidal situation, see <cit.>.


Let α<k-1. Then the roots of P^†(k,·) of valuation α are precisely the roots of P(k,·) of valuation α. Consequently, if m≤ k-1 then

    P(k,t)≡ P^†(k,t)  p^m.





For p≥ 5, the algorithm of Lauder presented in <cit.> computes P^†(k,t)  p^m for given k and m  and runs in polynomial time with respect to p, N and m, and linear time in log(k). This algorithm was extended by Vonk in <cit.> to include the primes p=2,3. These algorithms give us the input for our subsequent computations. 







§ ℒ-INVARIANTS AND DERIVATIVES OF THE CHARACTERISTIC SERIES OF U_P


For a fixed positive even integer k, in <cit.> Coleman, Stevens and Teitelbaum showed that whenever there is a unique split multiplicative cusp form f in S_k(Γ_0(pN)), it is possible to compute its ℒ-invariant as   

    Ł_p(f) = -2a_p(f)∂_κ P^†(k,t)∂_t P^†(k,t)|_t=a_p(f)^-1

where a_p(f) is the p­-th coefficient of the q-expansion of the newform. Clearly, the formula above works only in the special case where the space of split multiplicative forms in S_k(Γ_0(pN)) is 1-dimensional, as otherwise the denominator has a zero at t=a_p(f)^-1.  In <cit.>, Lauder uses the formula in this special case to compute the ℒ-invariant. 

The aim of this section is to remove the restriction on the dimension. If we denote by 𝒮_k(Γ_0(pN))^p-new the subspace of cuspforms which are new at p, then our approach can be described as follows. First we split 𝒮_k(Γ_0(pN))^p-new  into eigenspaces for the Atkin-Lehner operator at p. Then use higher derivatives to distinguish among the zeros of P^†(k,t) which correspond to different eigenforms, i.e. to separate the p­-adic families passing through the different eigenforms.

We begin by recalling the definition of p-adic ℒ-invariants. Note that for each f∈𝒮_k(Γ_0(pN))^p-new, we have

    U_pf=-p^(k-2)/2W_pf,

where W_p denotes the Atkin-Lehner involution acting on 𝒮_k(Γ_0(pN))^p-new. Let f∈𝒮_k(Γ_0(pN))^p-new be an eigenform for the Hecke operators away from N. Then f has an associated  ℒ-invariant ℒ_p(f)∈_p. By a generalization of the Greenberg-Stevens formula, see <cit.>, ℒ_p(f) is given as follows. The eigensystem (away from N) attached to f defines a classical point on the eigencurve 𝒞_N of level Γ_0(p)∩Γ_1(N), see <cit.>. There is a p-adic Coleman family f_κ=∑_n=1^∞ a_n(κ)q^n through f (i.e. an irreducible component of a small affinoid neighbourhood of the point attached to f in 𝒞_N that maps isomorphically onto an open affinoid subdomain of the weight space) of constant slope since the weight map is étale at the point attached to f, see <cit.>. In particular, via this p-adic family, the rigid morphism a_p(κ) on 𝒞_N defines a morphism on the affinoid subdomain of the weight space. The ℒ-invariant attached to f is then given by

    ℒ_p(f)= -2dlog(a_p(κ))|_κ=k.

Let us point out that, for κ sufficiently close to k, the eigenvalue a_p(κ) in level Γ_0(p)∩Γ_1(N) appears in fact in level Γ_0(pN); i.e. on the space 𝒮_κ^†(Γ_0(pN)). The expositions in <cit.> and <cit.> provide a nice summary of the constructions that are relevant in our setting.




Before we can study the relation between ℒ-invariants and the characteristic series of U_p, we need some preparations.



Let ϵ be an eigenvalue of the Atkin-Lehner operator W_p acting on 𝒮_k(Γ_0(pN)). Let a^ϵ_p,k=-ϵ p^(k-2)/2. Then 

    𝒮_k(Γ_0(pN))^p-new, ϵ=𝒮_k(Γ_0(pN))^U_p=a^ϵ_p,k,

where 𝒮_k(Γ_0(pN))^p-new,ϵ denotes the subspace of 𝒮_k(Γ_0(pN))^p-new
on which W_p has eigenvalue ϵ, and 𝒮_k(Γ_0(pN))^U_p=a^ϵ_p,k is the subspace of 𝒮_k(Γ_0(pN)) where the U_p operator acts as multiplication by a^ϵ_p,k.


By definition 𝒮_k(Γ_0(pN))^p-new, ϵ⊆𝒮_k(Γ_0(pN))^U_p=a^ϵ_p,k. In order to show 
the other inclusion, let us first remark that if a cuspform in 𝒮_k(Γ_0(pN))^U_p=a^ϵ_p,k is new at p, then clearly it is an eigenform for W_p with eigenvalue ϵ. 
Let us now look at the space of p-oldforms 𝒮_k(Γ_0(pN))^p-old. Each eigenvalue of T_p acting on 𝒮_k(Γ_0(N)) gives rise to two U_p-eigenvalues on 𝒮_k(Γ_0(pN))^p-old and by definition all eigenvalues of p-oldforms arise in this way. Let α be a T_p-eigenvalue. The corresponding U_p-eigenvalues are the roots of x^2-α x+p^k-1. Now, by a deep theorem of Deligne, these U_p-eigenvalues have complex absolute value p^(k-1)/2, so they cannot be a^ϵ_p,k.


Let us remark  that 𝒮_k(Γ_0(pN))^p-new has a basis of eigenforms for the Hecke operators away from N. These split into two orbits with respect to the eigenvalues of the Atkin-Lehner involution. 

From now on we fix an eigenvalue ϵ of W_p and set  d_ϵ=𝒮_k(Γ_0(pN))^p-new, ϵ. 
Let f_1,…,f_d_ϵ denote the basis of 𝒮_k(Γ_0(pN))^p-new, ϵ. For each i∈{1,…,d_ϵ}, we denote by a_i(κ) the p-th coefficient of the Coleman family passing through f_i as above.


Let ϵ be an eigenvalue of the Atkin-Lehner operator W_p acting on 𝒮_k(Γ_0(pN))^p-new, then 

    P^†(k,t)=(1-a^ϵ_p,kt)^d_ϵC(k,t),
 where C(k, (a^ϵ_p,k)^-1)≠ 0.


For α <k-1, by Coleman classicality, Theorem <ref>, the set of roots of P^†(k,t) of slope α is the set of roots of P(k,t) of slope α. Therefore, Lemma <ref> implies the claim.







There exists a small neighbourhood V⊂𝒲 of k such that in 𝒪(V){{ t}} we have

    P^†(κ,t)=∏_i=1^d_ϵ(1-a_i(κ)t) · C(κ,t).

where the function C(κ,t) specializes to the corresponding function in Proposition <ref> for κ=k.


By  <cit.>, we find a small neighbourhood V of k such that in 𝒪(V){{ t}} we have

    P^†(κ,t)=F(κ,t)· G(κ,t),

where F(κ,t)∈𝒪(V)[t] whose roots are precisely the roots of P^†(κ,t) of valuation less or equal to k-1 (with the same multiplicities), F(κ,0)=1  and G(κ,t)∈𝒪(V){{ t}}. After possibly shrinking V, the Coleman families through the f_i do not intersect and a_i(κ)∈𝒪(V). Then, by construction, these are zeros of F(κ,t) whose multiplicities add up to d_ϵ. Now factoring F(κ,t) gives the result.


In the sequel, for ∙∈{κ, t}, let us denote by ∂_∙^n P^†(k,t) the n-th partial derivative of P^†(κ,t) with respect to ∙ at the point (k,t), viewed as an element in _p{{ t}}.
 

For n≤ d_ϵ and ∙∈{κ, t}, we have

    ∂_∙^n P^†(k,t)= (1-a_p, k^ϵt)^d_ϵ-n· C^(n)_∙(k,t)

where

    C^(n)_∙(k,(a^ϵ_p,k)^-1)=n !· C(k,(a^ϵ_p,k)^-1)∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n∂_∙(1-a_m_j(κ)t)|_κ=k, t=(a^ϵ_p,k)^-1.


By Proposition <ref>, we have

    P^†(κ,t)=∏_i=1^d_ϵ(1-a_i(κ)t) · C(κ,t)

on a small neighbourhood of k. Thus, the n-th partial derivative of P^† for ∙∈{κ, t} satisfies

    ∂_∙^n P^†(κ,t)= ∑_i_1+i_2+…+i_d_ϵ+1=nn
    i_1,i_2,…,i_d_ϵ+1·∏_j=1^d_ϵ∂_∙^i_j(1-a_j(κ)t) ·∂_∙^i_d_ϵ+1C(κ,t).

For n≤ d_ϵ, let us split the sum into Σ_≤ 1(κ,t) and Σ_>1(κ,t) as follows:

    Σ_≤ 1(κ,t)    = ∑_i_1+i_2+…+i_d_ϵ+1=n, 
     i_j ≤ 1  ∀  j n i_1,i_2,…,i_d_ϵ+1·∏_j=1^d_ϵ∂_∙^i_j(1-a_j(κ)t)·∂_∙^i_d_ϵ+1C(κ,t), 
    Σ_>1(κ,t)     = ∑_i_1+i_2+…+i_d_ϵ+1=n,
     i_j > 1  for some  j n 
    i_1,i_2,…,i_d_ϵ+1·∏_j=1^d_ϵ∂_∙^i_j(1-a_j(κ)t)·∂_∙^i_d_ϵ+1C(κ,t).

We claim that Σ_>1(k,t) = (1-a_p,k^ϵ t)^d_ϵ-n+1· C^(n)_>1(k,t) for some C^(n)_>1(k,t).
 
If i_ℓ>1 for some ℓ∈{1,…,d_ϵ+1},  we must have i_j=0 for more than d_ϵ-n+1 many j's, since all i_j are non-negative, their sum equal to n and the cardinality of the range is d_ϵ+1. Specializing to κ=k, this immediately implies the result.
We can proceed in a similar fashion to show that the terms with i_d_ϵ+1 = 1 in Σ_≤ 1 satisfy an analogous formula. Therefore, we are left with the term

    n !· C(κ,t)∑_i_1+…+i_d_ϵ=n,
     i_j ≤ 1  ∀  j  ∏_j=1^d_ϵ∂_∙^i_j(1-a_j(κ)t).

Let us observe that after separating the terms with i_j=0 at κ=k and evaluating at t=(a^ϵ_p,k)^-1 we obtain

    n !· C(k,t)· (1-a_p, k^ϵt)^d_ϵ-n∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n∂_∙(1-a_m_j(κ)t)|_κ=k.

Putting these terms together completes the proof. 



In the spirit of the above proposition, one could also analyse the behaviour of mixed derivatives with respect to κ and t. However, for our applications only the above derivatives are relevant.


Finally we can state the main result of this section:



Let ϵ be an eigenvalue of the Atkin-Lehner operator W_p acting on 𝒮_k(Γ_0(pN))^p-new, then the monic polynomial 𝒬_p,k^ϵ(x)=∑_n=0^d_ϵ c_n x^d_ϵ-n∈_p[x],
where

    c_n:=2^n (a_p, k^ϵ)^nd_ϵ n∂^n_κ P^†(κ,t)∂^n_tP^†(κ,t)|_κ=k, t=(a_p, k^ϵ)^-1,

satisfies

    𝒬_p,k^ϵ(x)=∏_i=1^d_ϵ(x-ℒ_p(f_i)).


Taking the ratio of the n-th partial derivatives of P^†(κ, t) and applying Corollary <ref> we obtain 

    ∂^n_κ P^†(κ,t)∂^n_tP^†(κ,t)|_κ=k, t=(a_p, k^ϵ)^-1   =∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n∂_κ(1-a_m_j(κ)t)∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n∂_t(1-a_m_j(κ)t)|_κ=k, t=(a_p, k^ϵ)^-1
       =∑_1≤ m_1<…<m_n≤ d_ϵ (-a_p, k^ϵ)^-n∏_j=1^n a'_m_j(k)∑_1≤ m_1<…<m_n≤ d_ϵ (-a_p, k^ϵ)^-n
       =d_ϵ n^-1(a_p, k^ϵ)^-n∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n a'_m_j(k)a_p, k^ϵ.

Multiplying both sides by 2^n (a_p, k^ϵ)^nd_ϵ n yields

    c_n   =2^n (a_p, k^ϵ)^nd_ϵ n∂^n_κ P^†(κ,t)∂^n_tP^†(κ,t)|_κ=k, t=(a_p, k^ϵ)^-1
       =∑_1≤ m_1<…<m_n≤ d_ϵ ∏_j=1^n (2dlog(a_m_j(κ))|_κ=k)

The last term is precisely the (d_ϵ -n)­-th coefficient of the polynomial

    ∏_i=1^d_ϵ(x-ℒ_p(f_i)). ,

which completes the proof. 




Note that in the above proof we used that C_t^(n)(k,(a^ϵ_p,k)^-1)≠ 0. The analogous statement for C_κ^(n)(k,(a^ϵ_p,k)^-1) is equivalent to  all of the ℒ-invariants ℒ(f_i) being non-zero. While all our computational results support this claim, we are not aware of a proof of this statement. However, in the special case where f_i is of weight 2 and corresponds to an elliptic curve, this follows from a deep result in transcendental number theory stating that the Tate parameter is transcendental, see <cit.>.




§ DIMENSION FORMULAE


In this section we show how to compute the dimension d_ϵ of the cuspform space 𝒮_k(Γ_0(pN))^p-new, ϵ which appears in Section <ref>. Let us recall that N is a positive integer coprime to p. As before, we will denote by W_p the p-th Atkin-Lehner involution and whenever W_p is an operator on a vector space V, we will denote by tr_VW_p its trace on V.


The trace of the Atkin-Lehner operator W_p satisfies 
 
    tr_𝒮_k(Γ_0(pN))^p-newW_p=tr_𝒮_k(Γ_0(pN))W_p.



The cuspform space 𝒮_k(Γ_0(pN)) decomposes into a p-new and a p-old component 
𝒮_k(Γ_0(pN))=𝒮_k(Γ_0(pN))^p-new⊕𝒮_k(Γ_0(pN))^p-old.
The statement is then equivalent to say that tr_𝒮_k(Γ_0(pN))^p-oldW_p=0 and this follows from 
<cit.>.


In the case of N squarefree, building on previous works of Yamauchi, Skoruppa and Zagier, Martin gave a formula <cit.> for the trace of the Atkin-Lehner operator W_p on 𝒮_k(Γ_0(pN)). In the simple case where N=q>3 is a prime and p>3, we have

    tr_𝒮_k(Γ_0(pq))W_p=1/2(-1)^k a(p,q) h((√(-p))) (1+(Δ_p/q))+δ_k,2,

where a(p,q) is 1,4 or 2 for p congruent to 1 or 5, 3, 7 modulo 8 respectively, h((√(-p))) denotes the class number of (√(-p)) and Δ_p its discriminant, the Legendre symbol for Δ_p and q is denoted by (Δ_p/q), and δ_k,2 is 1 if k=2 and 0 otherwise.

By simple algebraic manipulations, we have the following


Let ϵ be an eigenvalue of the Atkin-Lehner operator W_p and set d:=𝒮_k(Γ_0(pN))^p-new. Then

    d_ϵ=1/2(d+ϵ tr_𝒮_k(Γ_0(pN))W_p)



Let us remark that the dimension of 𝒮_k(Γ_0(pN))^p-new can be obtained by recursively computing the dimension of 𝒮_k(Γ_0(pN))^p-old.  In Table <ref> there are dimensions of 𝒮_k(Γ_0(pN))^p-new and of 𝒮_k(Γ_0(pN))^p-new, ϵ for some levels and weights, where, for simplicity, we set Tr:=tr_𝒮_k(Γ_0(pN))W_p.








§ COMPUTING 𝒬_P,K^Ε


The main difficulty in computing  𝒬_p,k^ϵ is given by the partial derivatives, as shown in Theorem <ref>. For the denominators, this is straightforward by Remark <ref>, since we only use formal differentiation with respect to t. However, the computation of the numerators is more involved. 

In <cit.> Lauder explains a method using finite differences for the first derivative. The aim of this section is to generalize this method to derivatives of higher order.
The study of finite differences in this context is due to Gouvêa and Mazur, see <cit.>.

Let us first observe that it is enough to work only at classical points since, by definition, we have

    ∂^n_κ P^†(k,t)=lim_m →∞∂
        ^n-1_κ P^†(k + (p-1)p^m, t) - ∂^n-1_κ P^†(k,t)(p-1)p^m.

The finite differences used to approximate ∂^n_κ P^†(k,t) are defined as follows. Let s_m=(p-1)p^m and set 

    ∂^n_κ P^†(k,t)_m := s_m^-n∑_j=0^n(-1)^jn  j P^†(k + (n-j)s_m,t).





We have 

    ∂^n_κ P^†(k,t)_m+1≡∂^n_κ P^†(k,t)_m  p^m+1.

Therefore, ∂^n_κ P^†(k,t)≡∂^n_κ P^†(k,t)_m  p^m+1.


Let P^†(k,t)=∑_i=0^∞ b_i(k)t^i. In the sequel we fix a choice of i and write b(k)=b_i(k). Then, for s_m=(p-1)p^m it is enough to show

    ∑_j=0^n(-1)^jn  j b(k + (n-j)ps_m)=p^n ∑_j=0^n(-1)^jn  j b(k + (n-j)s_m)  p^(n+1)(m+1)

We define the difference functions δ_ν(b,k) recursively by

    δ_1(b,k)   :=b(k+s_m)-b(k),
    δ_ν(b,k)   :=δ_ν-1(b,k+s_m)-δ_ν-1(b,k)   for ν≥ 2.

By <cit.>, we have δ_ν(b,k)≡ 0  p^ν(m+1). Thus, rewriting the above equation, it suffices to show that

    ∑_j=0^n(-1)^jn  j b(k + (n-j)ps_m)-p^n ∑_j=0^n(-1)^jn  j b(k + (n-j)s_m)

is a -linear combination of δ_ν(b,k) for ν≥ n+1. By definition, we have

    δ_ν(b,k)=∑_j=0^νν  j(-1)^j b(k+(ν-j)s_m).

Let X be a variable and write

    (X-1)^ν=∑_j=0^νν  j(-1)^j X^ν-j.

Then, after substituting X^l for b(k+ls_m), we just need to prove that the polynomial R(X)=(X^p-1)^n-p^n(X-1)^n is a -linear combination of (X-1)^ν for ν≥ n+1 or equivalently, that R(X) vanishes to order at least n+1 at X=1. Clearly, R(X) vanishes to order at least n at X=1. Moreover, we have

    R(X)/(X-1)^n=(X^p-1/X-1)^n-p^n=(1+X+…+X^p-1)^n-p^n.

and by specializing to X=1, we see that R(X) vanishes to order at least n+1. This concludes the proof of the first part of the theorem. The second part is an immediate consequence, since the higher derivatives of every function given by a power series are approximated by finite differences on a small neighbourhood.









§ COMPUTATIONS




 §.§ Overview of the code

The algorithm implemented computes the polynomial 𝒬_p,k^ϵ, defined in Section <ref>.
The input of the algorithm are a prime p, a positive integer N coprime to p, an even integer k and a positive integer m, the desired precision. 

The code works in a straightforward manner following the steps below:


  * Find the Atkin-Lehner decomposition of the space. We will decompose the space according to the eigenvalues of the Atkin-Lehner operator W_p, since we will work on each 𝒮_k(Γ_0(pN))^p-new, ϵ separately, where ϵ is one of such eigenvalues. In this step, we actually only compute the dimension d_ϵ of such eigenspaces. As shown in Section <ref>, this computation can be done via Corollary <ref>, Equation (1.6) in <cit.> and by recursion.

  * Compute evaluations of the reverse characteristic series of the U_p operator on overconvergent modular forms at the weights {k,k+s_m,…,k+d_ϵ s_m} to precision (d_ϵ+1) m+1, where s_m=(p-1)p^m. We performed this step using algorithms of Lauder <cit.> and Vonk <cit.>, see Remark <ref>. 

  * For each subspace 𝒮_k(Γ_0(pN))^p-new, ϵ and 1≤ n ≤ d_ϵ, compute ∂_∙^n P^†(k,t), the partial derivative with respect to ∙ of P^†(κ,t) at the point (k,t) viewed as an element in _p{{t}}. By Theorem <ref>, Step 2 provides all data needed for the derivative in κ, which is the relevant piece. 

  * Build the quotient 𝒬^ϵ_p,k of both derivatives, if possible. If the precision is too low, this step cannot be performed. In this case, or if the resulting precision of the coefficients c_n as in Theorem <ref> is less than m, restart Step 2 with m replaced by m+10.





Some examples of the output are presented in the next subsection. Building from the output of the algorithm, we analyzed the distribution of (the slopes of) ℒ-invariants for increasing weight. We tabled the results obtained in Subsection <ref> and we formulated some speculations, based on the data collected, in Subsection <ref>.


It should be pointed out that, once the reverse characteristic series of U_p is obtained, our algorithm is independent of the work of Lauder and Vonk, and can thus be combined with different methods of computing the series. The main bottleneck of our computations is the p-adic precision to which the coefficients c_n of 𝒬_p,k^ϵ have to be computed. As shown in Theorem <ref>, this precision depends on the dimension d_ϵ of the relevant space of modular forms. This leads to work with very large matrices of U_p on overconvergent modular forms in the algorithms of Lauder and Vonk, making even the computation of the characteristic series rather time-consuming.




 §.§ Examples

All of the following examples were computed using an Intel^®
Xeon^® E5-2650 v4 CPU processor with 512 GB of RAM memory.

Let us consider the space 𝒮_4(Γ_0(6)). We have

    𝒮_4(Γ_0(6))^p-new,ϵ > 0   for    (p,ε)∈{(2,1), (3, 1)}.

In both cases the space is one-dimensional, i.e. the polynomial 𝒬^ε_p,4(X)∈ℚ_p[X] is given by a linear factor with the ℒ-invariant of the unique newform in 𝒮_4(Γ_0(6))^p-new as a zero. Modulo 2^21, respectively  3^21, we have

    𝒬^+1_2,4(X)    = X + 94387·2 
       = X - (2 + 2^3 + 2^4 + 2^7 + 2^9 + 2^10 + 2^11 + 2^12 + 2^16 + 2^18 + 2^19 + 2^20), 
    𝒬^+1_3,4(X)    = X - 41502709 
       = X - (1 + 3^2 + 2·3^7 + 3^8 + 2·3^9 +2·3^13 + 2·3^14 + 2·3^15).

The latter example was also computed in <cit.> and in <cit.>. The duration of the above computations was 1.8 and 5.6 seconds respectively.


















































Let p=7, N=11 and k=2. The space 𝒮_2(Γ_0(77))^7-new,ϵ is two-dimensional for ε = +1 and three-dimensional for ε = -1. The coefficients of the following polynomials have been reduced modulo 7^21.

    𝒬^+1_7,2(X)    = X^2 + 526982521374955003· 7· X + 192454376115114681· 7^2 
    𝒬^-1_7,2(X)    = X^3 + 222104449450143105· 7· X^2 
          - 103688086480269397· 7^2· X - 39596022807935252 · 7^3

The computation required approximately 2 hours. The slopes can be computed to be [1_2] and [1_3] respectively.  

In this case, we have  𝒮_2(Γ_0(77))^7-new,ϵ= 𝒮_2(Γ_0(77))^new,ϵ for ϵ=± 1. The space 𝒮_2(Γ_0(77))^new,+1 is spanned by two rational newforms f_1 and f_2. These correspond to the isogeny classes of elliptic curves with Cremona labels  and  and their ℒ-invariants are given by

    ℒ_7(f_1)   =2·7 + 3· 7^2 + 4· 7^3 + 7^4 + 7^5 + 4· 7^7 + 7^9 + O(7^10),
    
    ℒ_7(f_2)   =4· 7 + 2· 7^2 + 6· 7^3 + 5· 7^4 + 6· 7^5 + 6· 7^6 + 3· 7^7 + 3· 7^8 + 7^9 + O(7^10).

The ℒ-invariants attached to elliptic curves can be computed, for example, by a build-in method in terms of Tate-parameters in . In the two cases above, one first has to apply a quadratic twist, as the curves do not have split-multiplicative reduction at 7. The space 𝒮_2(Γ_0(77))^new,-1 is spanned by a rational newform f and two newforms g_1 and g_2 with coefficient field K=(√(5)). The newform f corresponds to the isogeny class of elliptic curves with Cremona label , which has split multiplicative reduction at 7. Its ℒ-invariant is given by

    ℒ_7(f)=3· 7 + 7^2 + 2· 7^3 + 5· 7^4 + 4· 7^5 + 7^6 + 4· 7^7 + 7^8 + 3· 7^9 + O(7^10).

If we divide the polynomial 𝒬^-1_7,2(X) by (X-ℒ_7(f)), we thus obtain (modulo 7^11)

    (X-ℒ_7(g_1))· (X-ℒ_7(g_2))= X^2+ 225931960·7· X+
    21342634· 7^2.

This polynomial is in fact irreducible over _7, showing that the ℒ-invariant ℒ_7(g_1) generates the quadratic unramified extension of _7, and similarly for ℒ_7(g_2). We have computed the coefficient fields _g_1,7 and _g_2,7  and obtained that _g_i,7=_7(ℒ_7(g_i)) for i=1,2. This is a first example of forms of  higher level (N=11), where Chenevier's description (for N=1) of the local coefficient field, mentioned in the introduction, still holds.






 §.§ Observations collected



Given a positive even integer k, we denote by ν_ℒ^ϵ(k,p,N) the finite sequence of slopes of the p-adic ℒ-invariants attached to forms in 𝒮_k(Γ_0(pN))^p-new,ϵ ordered in decreasing order, where ϵ is an eigenvalue of the Atkin-Lehner operator W_p acting on 𝒮_k(Γ_0(pN))^p-new. Note that this differs from the tables in <cit.>, where the space is decomposed with respect to the Atkin-Lehner involution at N. 

The observations collected are based on the data and on the tables presented in the next section.


We begin with the case p=2, that was extensively studied in <cit.>. There, several relations between slopes of different level and weights were conjectured for levels N=3,5,7. Table <ref> provides data for the case N=1, which is not accessible by the methods in <cit.>. The observations lead us to the following analogous conjecture.
[p=2] 



  (a)For k∈ 2+4, k≥ 10 and ϵ∈{±1}, we have

    ν_ℒ^ϵ(k,2,1)=ν_ℒ^-ϵ(k+6,2,1).


  (b) For every even integer k, the final min{d_+1,d_-1} slopes in ν_ℒ^+(k,2,1) and ν_ℒ^-(k,2,1) agree.


Table <ref> verifies this statement up to k=70 computationally.  Our observations is in line with similar observations in <cit.>. The table also provides the missing data for <cit.> stating that the slopes appearing in level 7 are the union of (two copies) of the slopes in levels 1 and 3.

The case p=3 and N=1 is treated in Table <ref>. Again we observe relations between various different slopes that do not yet have any theoretical interpretation. We collect our observations in the following conjecture.
[p=3] 



  (a) For k∈ 2+6, k≥ 8 and ϵ∈{±1}, we have

    ν_ℒ^ϵ(k,3,1)=ν_ℒ^ϵ(k+4,3,1).


  (b) For every even integer k, the final min{d_+1,d_-1} slopes in ν_ℒ^+(k,3,1) and ν_ℒ^-(k,3,1) agree.


Interestingly, the numbers 4 and 6 from Conjecture <ref> (a) are essentially switched here: we have relations between k and k+4 when considering k  6. The same is true for the Atkin-Lehner sign, since here we always have relations between slopes of the same sign. While Conjecture <ref> is very similar to Conjecture <ref>, we should point out that here there are slopes that do not have (obvious) relations to other slopes. These are the slopes for k∈ 4+6. We still observe that for example the slopes for k=10 and k=16 only differ by -1. Similarly, the slopes for k=22 and k=28 are very similar. However, we are not able to formulate a precise conjecture.

The data in Tables <ref>, <ref> and <ref> shows various relations between slopes of different weights supporting conjectural connections between them, analogous to the conjectures presented above. 
Moreover, after possibly removing oldforms, the analogous observation as in part (b) of the Conjectures <ref> and <ref> is still present in the data presented in Tables <ref> and <ref>. We are able to say even more, namely that observation (b) is a purely local phenomenon. Let for example p=13 and k=10, the corresponding space of modular forms consists of two Galois orbits of dimensions 4 and 5. Each of the orbits makes up one of the two columns in Table <ref>. Thus, the pairs in observation (b) arise from different orbits. Moreover, within each orbit (aside from the first slope) all slopes are distinct.

The data computed for p=2 and N=3 matches with the one presented in <cit.> and we observe again a relation between the slopes, see <cit.>. In all cases, it is an interesting question to analyze the growth of the slopes with the weight k further. However, at this stage, the collected data is not sufficient to make precise claims. We observe that, in general, almost all slopes are negative, thus providing evidence for the conjectureral statement  that all ℒ-invariants are non-zero in Remark <ref>. It would also be interesting to separate the slopes with respect to mod p eigensystems. This is however not possible with our method. As this is only relevant for larger p and N, this question becomes more interesting once more data in these cases is available.






 §.§ Tables


We keep the notation of subsection <ref>. Note that in the case N>1 our tables can include oldforms. Their slopes are indicated by 
 blue font. 















































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