Exponents for the Equidistribution of Shears and Applications
Dubi Kelmer Alex Kontorovich



Kelmer is partially supported by NSF CAREER grant  DMS-1651563.
kelmer@bc.edu
Boston College, Boston, MA

Kontorovich is partially supported by
an NSF CAREER grant DMS-1455705, an NSF FRG grant DMS-1463940,  a BSF grant, a Simons Fellowship,  a von Neumann Fellowship at IAS, and the IAS's NSF grant DMS-1638352.
alex.kontorovich@rutgers.edu
Rutgers University, New Brunswick, NJ and Institute for Advanced Study, Princeton, NJ


Exponents for the Equidistribution of Shears]
Exponents for the Equidistribution of Shears and Applications



In <cit.>, the authors introduced “soft” methods to prove the effective (i.e. with power savings error) equidistribution of “shears” in cusped hyperbolic surfaces.

In this paper, we study the same problem but
 
now
 allow full use of the spectral theory of automorphic forms 
 
 to produce 
explicit
 exponents, 
 and
 uniformity in 
 parameters. 
We 
give applications to counting 
square values of   quadratic forms.


[
    Alex Kontorovich
    December 30, 2023
=====================







§ INTRODUCTION




 §.§ Equidistribution

Let <G:=_2() be a non-uniform lattice, and let 

    _0∈ T^1()≅ G
 
be a base point in the unit tangent bundle of the punctured surface , so that the visual limit point 

    =lim_t→∞_0· a_t

is a cusp of . Here a_t=(e^t/2,e^-t/2) is the geodesic flow; let 
A^+={a_t,t>0}. 
As in <cit.>, we define a shear of the cuspidal geodesic ray _0· A^+
to be its left-translate by 

    _T:=(T^2+1)^-1/4T(T^2+1)^-1/40(T^2+1)^1/4.

For example, identifying G/K with  under g↦ g · i (here K=(2) is a maximal compact subgroup) and taking  _0=e, the identity element of G, we have for a right-K-invariant test function Ψ∈ C_c( G)^K that its evaluation along such a shear is given by

    ∫_a∈ A^+Ψ(_0· a·_T)da = 
    ∫_1/√(T^2+1)
    ^∞Ψ(Ty+iy)dy y.

In <cit.>, the authors proved the effective equidistribution of  shears (as T→∞) using “soft” ergodic methods (e.g. mixing) and basic properties of Eisenstein series.[See also <cit.>, where an asymptotic formula is obtained by different means, with an  error term weaker than power savings.]
The goal of this paper is to use more of the  spectral theory of automorphic forms 
to produce explicit 
exponents in this problem.

For ease of exposition, and to write the best exponents that come from our method, we restrict below to  conjugate to the congruence group _0(p).
A special case of t:Equidistribution1 below gives the following  bound.


Assume the Ramanujan conjecture for the exponent bounding the Fourier coefficients of Maass forms on  (see sec:3). 
Then for any Ψ∈ C^∞_c( G)^K, any T≥2, and any >0, there are constants C_j=C_j(Ψ), j=1,2, so that

    ∫_a∈ A^+Ψ(_0· a·_T)da = 
    C_1log T + C_2 + O_Ψ,(T^-1/4+).





We take the opportunity here to correct an error in the analysis in <cit.>, which has no effect on the qualitative power gain, but does affect the exponents as explicitly quantified here. In particular, <cit.> is incorrect as stated, and we do not know how to obtain square-root cancellation by this approach. See rmk:error for the error and how to correct it.




 §.§ Counting

As is standard, such equidistribution results can be applied to counting problems in discrete orbits. In particular, see <cit.> where we explain that proving the effictive equidistribution of shears settles the remaining lacunary cases of the Duke-Rudnick-Sarnak/Eskin-McMullen program <cit.> of effectively counting discrete orbits on quadrics in archimedean balls. 
In smooth form, one can produce from thm:1 above some 
rather sharp error exponents, as we now illustrate.

Let F be a real ternary indefinite quadratic form, let G=_F^∘() be the connected component of the real special orthogonal group preserving F, and assume that <G is the image of _0(p) under a spin morphism _2()→_F^∘() (see sec:spin).
Let ψ:G→_≥0 be a smooth bump function about a sufficiently small bi-K-invariant neighborhood of the identity in G (that is, a region of the form (<ref>)), with ∫_Gψ=1. 

Fix _0∈^3 so that the orbit 

    :=_0· ⊂ ^3
 
is discrete, the stabilizer of _0 in G is a split torus, H, say, and ∩ H is finite. 
Fix a right-K-invariant archimedean norm · on H G≅_0· G. This data  induces a smoothed indicator function of a norm-T ball on ∈_0· G via convolution with ψ:

ψ_T():=∫_g∈G_{g<T}ψ(g)dg.

In sec:5 (see prop:smoothC), we prove the following

Again assume the Ramanujan conjecture for Maass forms on .
Then for any >0, 
there are constants C_j=C_j(ψ), j=1,2, so that 


    ∑_∈ψ_T( ) = C_1 Tlog T + C_2 T +O_,ψ(T^34+).

Unconditionally, the error exponent 3/4+ can be replaced by 3+4/4+8+, where =7/64 is the best currently known bound towards the Ramanujan Conjecture (which stipulates that =0 holds).




 §.§ Explicit Constants


In certain settings of classical interest, one can go a step further and explicitly identify the constants  C_j appearing in the main terms above. 

To 



showcase
this fact, we



count
 integer points on the inhomogeneous Pythagorean quadric:

    W_d : x^2+y^2-z^2 = d,

when d is a perfect square (which corresponds to ∩ H finite as above). 
After unsmoothing the count in thm:2 to make the constants independent of the smoothing function ψ, we obtain the following.

Let d=n^2 be a square, with n∈_>0. 
Define the counting function

    _d(T)  := # W_d()∩{x^2+y^2+z^2<T^2}
    .

Again let =7/64 be the  bound towards the Ramanujan Conjecture (that θ=0).
For any η<3/40+72θ, β>3/2+2θ, 
and T≥ d^, 
 we have that

_d(T)
 = _d(T)+
O(T^1-ηd^βη)
,
where the “main term” is given by

    _d(T) :=  √(128)T/π(log(T)+C
    -D(n)+log(2)(13-12^ν+2)
    )
    .

Here
 ν=ν_2(n) is the 2-adic valuation of n,
the constant  C is given by:

    C
    
    
       =    2-1
    -2'(2)-log(2)2-log(|(1/4)|^44π^3)
     = 0.616174…,

where =0.577… is Euler's constant, ζ(s) is the Riemann zeta function,  (s) is the Gamma function, and ϕ is the Euler totient function,
and






the 
factor
 D(n) is
 the Dirichlet coefficient of (^2·')(s-1), that is,


    D(n)=1/n∑_a|nϕ(na)log(a).





To illustrate the validity of this complicated formula, we verify it numerically with plots of _d(T), _d(T), and their difference, for d=144; see fig:1. For T as large as 10,000, the counting function reaches around 350,000, while the difference _d(T)-_d(T) remains of size around 400≍√(T), suggesting perhaps that (<ref>) may remain valid with any η<1/2 and =1/2.


Our results are meaningful as long as T>d^3/2+θ+ϵ. In particular, assuming Ramanujan we can take T to be almost as small as d^3/2. When T<√(d) we trivially have _d(T)=0 and it is an interesting problem to obtain meaningful asymptotics also for the range √(d)<T<d^3/2. (In the rather different setting of d  being a fundamental discriminant, Friedlander-Iwaniec <cit.>, using different tools showed an asymptotic formula which is effective for T almost as small as √(d).)




































 
 
 Along the way to proving thm:3, we need to establish the following result,  counting the number of binary quadratic forms of a fixed square discriminant d with coefficients in a norm ball. As it is possibly of independent interest,  we record it here.
Let  

    Q(a,b,c)=b^2-4ac,
 
and

    _Q,d(T)={(a,b,c)∈^3:Q(a,b,c)=d,  2a^2+b^2+2c^2≤ T^2}.



With all notation and assumptions as in thm:3, we have

_Q,d(T)=√(72)T/π(log(T)+C-D(n)+O(d^ηβ/T^η)).


As discussed in <cit.>, there are 
many other methods for counting such expressions. For example, Hulse et al <cit.> used a Multiple Dirichlet Series technique to count binary forms with a fixed discriminant. In smooth form and counting in a slightly different region, they obtain a version of (<ref>) with square-root error in the T aspect, but with no visible uniformity in d. It is also not clear how easy it would be to convert their constants into the completely numerically explicit values as in (<ref>).





 §.§ Notation

Throughout this paper we denote by G=_2() and  a non-uniform lattice in G. For ease of exposition (and for our applications), we assume that  is conjugate to _0(p); minor modifications are needed to handle a more general setting. 
We will use the notation A(t)≪ B(t) to mean that there is some constant c>0 such that A(t)≤ cB(t), and we will use subscripts to indicate the dependance of the constant on parameters. For B(t)≥ 0 the notation A(t)=O(B(t)) means that |A(t)|≪ B(t). 
We write A(t)≍ B(t) for A(t)≪ B(t) ≪ A(t).



 §.§ Organization

The preliminary sec:prelim
reviews spectral decompositions, Eisenstein series, and Sobolev norms.
Then in sec:3, we improve on our analysis in <cit.>, giving stronger estimates for Fourier expansions at various cusps in terms of approximations to the Ramanujan Conjectures. We use these, together with some slight improvements on the method in <cit.>, to prove  in sec:edShears the sharp equidistribution of shears claimed in thm:1. The counting thm:2 is derived from this in sec:5, and then used in sec:6 to prove the explicit counting thm:3 and <ref>.
Calculations of Fourier expansions for Eisenstein series on _0(p) are reserved for the Appendix.





 §.§ Acknowledgements


We thank Zeev Rudnick for comments on an earlier draft.





§ PRELIMINARIES



 §.§ Coordinates

Let K,A,N≤ G denote the orthogonal group, the diagonal group, and the unipotent group respectively.
Explicitly let 

    k_θ=([  cosθ  sinθ; -sinθ  cosθ ]),  a_y=([  y^1/2      0;      0 y^-1/2 ]), n_x=([ 1 x; 0 1 ]),
 parametrize elements in K,A and N respectively.
The decomposition G=NAK gives coordinates g=n_xa_yk_θ on G, and the Haar measure in these coordinates is

    dg=dxdydθ/2π y^2.

The group G acts on the upper half space  ={z∈: (z)>0} by linear fractional transformations, explicitly, for g=([ a b; c d ]), gz=az+b/cz+d preserving the hyperbolic area dμ=dxdy/y^2.

For any lattice  we identify the quotient  with G/K (in particular, we will think of functions on  as right K-invariant functions on G). 
The hyperbolic area v_=μ() is the same as the Haar measure of G.

For _1=_2() we recall that v__1=π/3.



 §.§ Sobolev Norms

Fix a basis ={X_1,X_2,X_3} for the Lie algebra 𝔤 of G, and given a smooth test function Ψ∈ C^∞( G), define the “L^p, order-d” Sobolev norm _p,d(Ψ)  as

    _p,d(Ψ)  : =  ∑_()≤ dΨ_L^p( G)
    .

Here  ranges over monomials in  of order at most d. Note that since the right action of  commutes with the left action of G, all norms are invariant in the sense that _p,d(Ψ^τ)=_p,d(Ψ) where Ψ^τ(x)=Ψ(τ x).

We will work with various norms that are convex combinations of these Soboev norms and it will be convinient to classify these norms with respect to how large they 
become
on functions approximating a small bump function. For small δ>0 let 

    B_δ=KA_δ K

denote a (spherical) δ-neighborhood of the identity,
 where 
 
    A_δ={a_y: |log(y)|<δ} .
 
 
 We say that a norm  is of degree α if one can construct a family of smooth functions ψ_δ on K G/K, supported on B_δ, with average 
 
    ∫_G ψ_δ(g)dg=1,
 
 so that the corresponding periodized functions 
 
    Ψ_δ(g)=∑_∈ψ_( g),
 
 have norms growing like (Ψ_δ)≍δ^-α. 
 We will slightly abuse notation and sometimes denote by _α a norm of degree α, without specifying 

the norm explicitly.
 
Note for future reference that the  
Sobolev norm _p,d is of degree d+2-2/p, and that if _α and _β are of degrees α and β respectively, then the convex combination _α^q_β^1-q is of degree qα+(1-q)β. 
In particular, the L^2 norm has degree 1, while the L^∞ norm has degree 2.




 §.§ Eisenstein series

For any cusp  of  let _ denote the stabilizer of  in  and τ_∈ G be a corresponding scaling matrix such that τ_∞= and

    τ_^-1τ_∩ N={n_k:k∈}.

In particular, for the congruence groups _0(p) there are two cusps, one at ∞ of 
width 
1, and the other at 0 of width p with scaling matrix τ_0=([      0 1/√(p);  -√(p)      0 ])=k_π/2· a_p.

The Eisenstein series corresponding to a cusp  is defined for (s)>1 by

    E_(z,s)=∑_∈_((τ_^-1 z))^s

and has a meromorphic continuation to  with a simple pole at s=1 with residue 1/v_ and (since  is congruence) no other poles in (s)>. 








One can regularize the Eisenstein series by subtracting the pole at s=1, and we define the corresponding Kronecker limit by

    _,(z)=lim_s→ 1(E_,(s,z)-1/v_(s-1)).











































































 §.§ Spectral decomposition

The hyperbolic Laplace operator  
acts (after unique extension) on the space
  L^2() of square-integrable automorphic functions, and is self-adjoint and positive semi-definite.
The spectrum of  is composed of the constant functions, the continuous part (spanned by Eisenstein series), and discrete part (spanned, in the congruence case, by 
Maass cusp forms). 







We denote by () the space spanned by the Eisenstein series
and by () its orthogonal complement which is the space of cusp forms.
For =_0(p) the space of cusp forms further decomposes into the space of old forms _ old() spanned by the set
 {(z),(pz): ∈(_1)}, and its orthogonal complement _ new(). 
 
For congruence , the space of cusp forms has an orthonormal basis composed of Hecke-Maass forms, that are joint eigenfunctions of the Laplacian and all Hecke operators.
We have the following spectral decomposition (see <cit.>).

For Ψ∈ L^2()

    Ψ(z)   =   μ_(Ψ)+∑_kΨ,_k_k(z)
          +∑_1/4π∫_-∞^∞Ψ,E_,(·,12+ir)E_,(z,12+ir)dr
    ,

where μ_(Ψ)=1/v_∫_Ψ(z)dμ(z) and the first sum is over an orthonormal basis of 
Maass cusp forms.


The equality is in L^2() and pointwise for Ψ∈ C^∞_c(). As a direct consequence, we have the following

For Ψ∈ C^∞_c()

    Ψ^2   =   |μ_(Ψ)|^2+∑_k|Ψ,_k|^2+
          ∑_1/4π∫_-∞^∞ |Ψ,E_,(·,12+ir)|^2dr.

























 










§ FOURIER COEFFICIENTS


In this section, we derive general bounds for Fourier coefficients of test functions at various cusps. In principle, most of the steps are standard, but we did not find a reference in the literature which carries out each of the necessary calculations, so we give details for the reader's benefit. Another reason for restricting to  conjugate to _0(p) is that the general theory of Fourier coefficients at arbitrary cusps becomes extremely cumbersome (see, e.g., <cit.>).

First we specify what we mean by the “Ramanujan conjectures.”
Let  be a Hecke-Maass cusp newform for _0(p), with Laplace Eigenvalue 14+r^2. 
For m≠0, its mth Fourier coefficient (at the cusp =∞) satisfies

    a_,∞(m,y):=
    ∫_0^1
    (x+iy)e(-mx)dx
    =a_,∞(m)
    √(y)K_
    ir
    (2π |m| y),

with K_s(y) the Bessel function of the second kind.
The coefficient  further decomposes as

    a_,∞(m)=a_,∞(1)λ(m),

where
 λ(m) the corresponding Hecke eigenvalue. Let ∈[0,1/2] be a number so that

|(m)| ≪_|m|^+.
In particular,

    θ=7/64

is known <cit.>, while 
 the Ramanujan conjecture predicts that θ=0 holds.


Now, let  be a cusp  of a lattice , 
and
let τ_ denote the corresponding scaling matrix.
Then for any test function Ψ∈ C^∞_c(), the 


translated
function
Ψ^τ_(z):=Ψ(τ_ z) is periodic in x with period one and hence has a Fourier expansion

    Ψ^τ_(z)=∑_m∈a_Ψ,(m;y)e^2π i mx.


In <cit.>, we 
proved
that there are constants 0<c_<∞ and 0<η_<1 and some norm  (a convex combination of Sobolev norms) such that these coefficients satisfy 

    |a_Ψ,(m,y)|≪_(Ψ)|m|^c_y^η_

uniformly for all 0≠ m∈ and y>0. The argument there was quite soft (using mixing) and applied to any lattice. 
Now we specialize to  conjugate to _0(p)


to improve the exponents c_ and η_ above, as well as to have better control on the degree of the Sobolev norm . 
Our main result is the following.

Let  be conjugate to 
_0(p).
For any Ψ∈ C^∞_c(), for any cusp  of  we have that

    a_Ψ,(0,y)=μ_(Ψ)+O(Ψ_2^3/4Ψ_2^1/4y^1/2).

Moreover, for any m≠0, for any ϵ>0 and  any 

    α_0>5/3,
 
we have

    |a_Ψ,(m,y)|≪_α_0,ϵ,p_α_0(Ψ)y^12-ϵ|m|^θ+ϵ,

where
_α_0 is a norm of degree α_0.


In order to prove  p:Fourier we consider the spectral decomposition of Ψ into Maass forms and Eisenstein series and bound the Fourier coefficients of each. Explicitly, for the cusp forms we show the following.

Let _k be a Hecke-Maass cusp form on _0(p) with eigenvalue 14+r_k^2. Then for  any cusp , any m≠0, and any ϵ>0, we have

    |a__k,(m,y)|≪_ϵ,p  (r_k+1)^-1/3+ϵy^1/2-ϵ|m|^θ+ϵ

 

When  is an eigenfunction of the Laplacian with eigenvalue s(1-s) we have that
a_,(0,y) is a linear combination of y^s and y^1-s and for m≠ 0 it takes the form

    a_Ψ,(m;y)=a_Ψ,(m)√(y)K_s-1/2(2π m y),

with K_s(y) the Bessel function of the second kind.


Recall that _0(p) has two inequivalent cusps, one at ∞ and one at 0. First assume that  is equivalent to ∞.
Combining (<ref>) with  Hoffstein-Lockhart's  <cit.> control on the size of a_,∞(1),
we obtain the bound

    |a_,∞(m)|≪_ϵ (|m|r)^ϵ |m|^θ e^π r/2.


Together with the bound <cit.>

for the Bessel function,

    |K_ir(y)|≪_ϵ e^-π r/2(r+1)^-1/3+ϵy^-ϵmin{1,e^π r/2-y},

we see that (<ref>) holds in this case. 


For the cusp at 0 we note that the scaling matrix τ_0=([      0 1/√(p);  -√(p)      0 ]) commutes with the Hecke operators T(n) with (n,p)=1 and satisfies that 
τ_0^-1_0(p)τ_0=_0(p) (see  <cit.>). Hence _k^τ_0 is also Hecke eigenfunction with the same eigenvalues, and from multiplicity one for new forms  we get that 
_k^τ_0=c_k with some scalar c of modulus 1. Hence, in absolute value, |a__k,∞(m)|=|a__k,0(m)| so we have the same bound also for the cusp at 0.

Finally, the bound for old forms follows directly from the bound for new forms of _1=_2(). Explicitly, let  be a Hecke-Maass form for _1 with  Fourier coefficients a_(m,y). From this form we get two companion forms _1(z)=(z) and _2(z)=(pz) invariant under _0(p). For the cusp at infinity _1= has the same Fourier expansion at infinity as . For the second form 

    _2(z)=(p z)=∑_m a_(m,py)e^2π im px=∑_p|m a_(mp,py)e^2π imx,

hence  a__2,∞(m,y)=a_(m/p,py) if p|m and is zero otherwise.
The cusp at zero has scaling matrix τ_0=k_π/2· a_p. Write =k_π/2 so that τ_0= a_p. Since ^σ= we get that 
_1^τ_0=^σ a_p=^a_p=_2, whence a__1,0=a__2,∞. Similarly
_2^τ_0=^a_pσ a_p=^σ= and a__2,0=a__1,∞. Thus the same bound holds also for the cusp at zero.




Next, we need to bound the Fourier coefficients of Eisenstein series. For each pair of cusps , of  the Fourier expansion of the Eisenstein series E_, with respect to the cusp at , is given by

    E^τ__,(z,s)=δ_,y^s+ϕ_,(s)y^1-s+∑_m≠ 0a_,(s;m,y)e(mx).


For  conjugate to _0(p) and any two cusps ,, we have 

    |a_,(12+ir;m,y)|≪_ϵ y^1/2-ϵ|m|^ϵ (1+|r|)^-1/3+ϵ.



Since E_,(z,s) is an Eigenfunction with eigenvalue s(1-s) we can write

    a_,(s;m,y)=ϕ_,(s;m)2√(y)K_s-1/2(2π my).

For the full modular group _1=_2() there is just one cusp at ∞ and the Fourier coefficients are given explicitly by
ϕ(s)=ζ^*(2s-1)/ζ^*(2s) and

    ϕ(s,m)=τ_s-1/2(m)/ζ^*(2s),

where ζ^*(s)=π^-s/2ζ(s)(s/2) is the completed Riemann zeta function and  τ_s(m)=∑_ab=|m|( ab)^s is the divisor function 
<cit.>. 
In particular, using the Stirling approximation for the -function and (<ref>) for the Bessel function gives (<ref>) in this case.
For the congruence groups _0(p) the coefficients ϕ_,(s,m) are given by a similar explicit formula (see p:phiEis below), resulting in the same bound.
 

Combining  the above bounds for Fourier coefficients of Maass forms and Eisenstein series we can use the spectral decomposition to bound the Fourier coefficients of any smooth function as follows.

First, noting that  Ψ∈ C^∞_c(_0(p)) iff Ψ^τ∈ C^∞_c() and the Fourier coefficients satisfy 
|a_Ψ,(m,y)|=|a_Ψ^τ,(m,y)| with =τ^-1, we may assume that =_0(p).

Let =_0(p) and Ψ∈ C^∞_c(). Using the spectral expansion we can write (for any cusp ) and m≠ 0

    a_Ψ,(m,y)   =   ∑_k ⟨Ψ,_k⟩ a__k,(m,y)
          +∑_1/2π∫_⟨Ψ,E_,(·,12+ir)⟩ a_,(12+ir;m,y)dr
    .

To bound the contribution of the first sum fix a large parameter M (to be determined later). Applying the bound (<ref>) to the Fourier coefficients
we get 

    |∑_k ⟨Ψ,_k⟩ a__k,(m,y)|   ≪_ϵ   
    y^1/2-ϵm^θ+ϵ(∑_r_k≤ M|⟨Ψ,_k⟩|/(r_k+1)^1/3-ϵ
          1in
    +∑_r_k≥ M|⟨Ψ,_k⟩|/r_k^7/3-ϵ)
    .

Using Cauchy-Schwarz, and Weyl's law stating that #{r_k≤ M}≪ M^2 we can bound 

the first sum  by

    Ψ_2_2√(∑_r_k≤ M(r_k+1)^-2/3+ϵ)≪ M^2/3+ϵ_2,0(Ψ),

and the second by

    Ψ_2√(∑_r_k> M r_k^-14/3+ϵ)≪ M^-4/3+ϵ_2,2(Ψ).

Choosing M=_2,0^-1/2_2,2^1/2 we get that 

    |∑_k ⟨Ψ,_k⟩ a__k,(m,y)|≪_ϵ y^1/2-ϵm^θ+ϵ_53+ϵ(Ψ),

where the norm _5/3+2ϵ(Ψ)=_2,0(Ψ)^2/3-ϵ/2_2,2(Ψ)^1/3+ϵ/2 is of degree 5/3+ϵ.

Next for the Eisenstein integrals for each pair of cusps , use (<ref>) to get



    |∫_⟨Ψ,E(·,12+ir)⟩ a_,(12+ir;m,y)dr|   ≪_ϵ   
     y^1/2-ϵm^ϵ(∫_|r|≤ M|⟨Ψ,E(·,12+ir)⟩|/(1+r)^1/3-ϵdr
          .5in
    + ∫_|r|> M|⟨Ψ,E(·,12+ir)⟩|/r^7/3-ϵ)dr
    .

As before we can use Cauchy-Schwarz to bound the first integral by 
O_ϵ(_2,0(Ψ)M^1/6+ϵ) and the second by O_ϵ(_2,2(Ψ)M^-11/6+ϵ) so taking M=_2,0^-1/2_2,2^1/2 the whole integral is bounded by

    ∫_⟨Ψ,E(·,12+ir)⟩ a_r(m,y)dr≪ y^1/2-ϵm^ϵ_7/6+ϵ(Ψ)
    ,

where 

    _7/6+ϵ(Ψ)=_2,0(Ψ)^11/1-ϵ/2_2,2^1/12+ϵ/2
    ,

is of degree 7/6+ϵ.

Collecting the contributions of all cusps we get that 

    |a_Ψ(m,y)|≪ y^1/2-ϵm^ϵ(m^θ_53+2ϵ(Ψ)+_14/12+2ϵ(Ψ)).

Taking ϵ sufficiently small so that 53+2ϵ=α_0 concludes the proof for the non trivial coefficients.

For the trivial (m=0) coefficient, again using the spectral expansion, the only contribution comes from the constant function (giving the main term) and the Eisenstein integrals. We thus need to bound 
for each pair of cusps

    ∫_Ψ,E_,(12+ir,·)(y^1/2+ir+ϕ_,(12+ir)y^1/2-ir)dr
    ≪ y^1/2∫_|Ψ,E_,(12+ir,·)|dr.

As before we separate 

    ∫_|Ψ,E_(12+ir,·)|dr =
    ∫_|r|<M|Ψ,E_,(12+ir,·)|dr
     +∫_|r|>M|Ψ,E_,(12+ir,·)|dr.

Using Cauchy Schwarz, we can bound the first integral by

    ∫_|r|<M|Ψ,E_,(12+ir,·)|dr   ≤   √(2M)√(∫_|Ψ,E_,(12+ir,·)|^2dr)
       ≪   √(M)Ψ_2.

and

    ∫_|r|>M|Ψ,E_,(12+ir,·)|dr   =   ∫_|r|>M| f,E_,(12+ir,·)|/1+r^2dr
    
       ≪   Ψ_2 M^-3/2,

We thus get that

    ∫_|Ψ,E_,(12+ir,·)|dr≪√(M)Ψ_2+ M^-3/2Ψ_2,

and the optimal choice of M=√( f_2/f_2) 
gives the desired bound.




 §.§ Additonal estimates

We end this section with several estimates which follow from our bounds on Fourier coefficients. 
First, as consequence of the estimate for a_Ψ,(0,y) we get the following useful estimate that we record here for future use.

Let  be conjugate to _0(p). For any cusp  with scaling matrix τ_, for any ψ∈ C^∞_c() we have the bound

    (∫_0^1|ψ^τ_(x+iy)|^2dx)^1/2≪_1(ψ)+_2(ψ)y^1/4,

with _1 and _2 are suitable norms of degree 1 and 2 respectively.


When y is small and ψ approximates a bump function this is an improvement over the trivial bound of _∞,0(ψ) which is a norm of degree 2.


Using  (<ref>) with the test function Ψ(z)=|ψ^τ_(z)|^2, we get that

    ∫_0^1|ψ^τ_(x+iy)|^2dx=μ(Ψ)+O(Ψ_2^3/4Ψ_2^1/4y^1/2)).

For Ψ=|ψ|^2 we have

    μ(Ψ)=|_2,0(ψ)|^2, Ψ_2= |_4,0(ψ)|^2, Ψ_2≪ |_4,2(ψ)|^2.

Define the norms _1(ψ)=_2,0(ψ) and

    _2(ψ):=_4,0(ψ)^3/4_4,2(ψ)^1/4.

Clearly _1 is of degree 1 and _2 is of  degree 2. 
Finally taking a square root gives the result.



Combining c:L2bound and p:Fourier we obtain another estimate that we will need.

Let  be conjugate to _0(p) and Ψ∈_c^∞(). Then for any α_1>5+6θ/3+6θ and any η_1<1/2+4θ

    ∑_m≠ 0|a_Ψ,(m,y)|/m≪_α_1,η_1_α_1(Ψ)y^η_1+_α_1+1(Ψ)y^η_1+1/4,

with _α_1 and _α_1+1 appropriate norms of degrees α_1 and α_1+1 respectively.


Replacing Ψ with Ψ^τ_ we may assume that =∞ and  has cusp of width one. 
We can find ϵ>0 and α_0>5/3, sufficiently small so that α_1=α_0+2θ+2ϵ/1+2θ+2ϵ and that
η_1=1+2ϵ/1+2θ-2ϵ.
Fix a large parameter M and separate the sum to 

    ∑_m≠ 0|a_Ψ,(m,y)|/m=∑_|m|≤ M|a_Ψ,(m,y)|/m+∑_m>M|a_Ψ,(m,y)|/m

For the first sum, using  p:Fourier we get that 
   
    ∑_|m|≤ M|a_Ψ,(m,y)|/m≪_α_0(Ψ)y^1/2-ϵM^θ+ϵ.

For the second sum using Cauchy Schwarz, followed by Parseval, and then applying  c:L2bound we get

    ∑_m>M|a_Ψ,(m,y)|/m   ≤   1/√(M)(∑_|m|>M|a_Ψ,(m,y)|^2)^1/2
       ≤   1/√(M)(∫_0^1 |Ψ(x+iy)|^2dx )^1/2
       ≪   _1(Ψ)+_2(Ψ)y^1/4/√(M).

Fix an optimal choice of M=(_1(Ψ)/_2(Ψ))^2/1+2θ+ϵy^2ϵ-1/1+2θ+2ϵ to get that

    ∑_m≠ 0|a_Ψ,(m,y)|/m≪_α_1(Ψ)y^η_1+_α_1+1(Ψ)y^η_1+1/4,

with the norms _α_1(Ψ)=_1(Ψ)^2θ+2ϵ/1+2θ+2ϵ_α_0(Ψ)^1/1+2θ+2ϵ and _α_1+1(Ψ)= _α_1(Ψ)_2(Ψ)_1(Ψ).







§ EQUIDISTRIBUTION OF SHEARS

We now  use our results on the Fourier coefficients from the previous section to improve the error term in the equidistribution result of <cit.>.
In addition to improving the error term we also take care to make the dependance of the error on width of the cusp explicit, as this will be needed for our application.
We will show the following

Let  be a conjugate of _0(p) with a cusp at ∞ of width ω≥ 1.
For any 7+12θ3+6θ<α<3 and any 14<η_1<12+4θ, there are norms _α,_α_2 of degrees α and α_2=1+α+3-α4η respectively , so that for any Ψ∈ C_c^∞() and any T≥ω^1/η_1

    ∫_1/√(1+T^2)^∞Ψ(yT+iy)dy/y   =   μ_(Ψ)log(Tω )+⟨_,∞,Ψ⟩
          +O_α,η_1(_α(Ψ)ω^1/2/T^η_1/2+_α_2(Ψ)ω^1/4/T^η_1/2+1/8)
    
    .




As in <cit.>, the proof of  t:Equidistribution1 splits into two parts. The first gives equidistribution in the strip (compare to <cit.>) which in our setting is


Under the same conditions and notations as in  t:Equidistribution1 we have




    ∫_1/√(1+T^2)^∞Ψ(yT+iy)dy/y   =   1/ω∫_0^ω∫_1/T^∞Ψ(x+iy)dydx/y
          +O_α,η_1(_α(Ψ)ω^1/2/T^η_1/2+_α_2(Ψ)ω^1/4/T^η_1/2+1/8).



The second step  uses the theory of Eisenstein series to estimate the strip average by the Eisenstein distribution (see <cit.>). 

Under the same conditions and notations as in  t:Equidistribution1 we have, 

    1/ω∫_0^ω∫_1/T^∞Ψ(x+iy)dydx/y   =   μ_(Ψ)log(Tω)+⟨_,∞,Ψ⟩+O(_2,0(Ψ)/√(ω T))
    .




The proof of  p:Eisenstein follows exactly as that of <cit.> without any changes. The only difference is the absence of residual spectrum in this case, and the fact that we are considering K-invariant test functions, allowing us to use the Sobolev norm _2,0(Ψ) instead of _2,1(Ψ).  t:Equidistribution1 follows from these two propositions after noting that the error term O(_2,0(Ψ)/√(ω T)) is subsumed by the other terms.
We thus devote the rest of this section to the proof of  p:Strip, taking advantage of the assumption that
 
  is a conjugate of the congruence group _0(p). 






 §.§ Equidistribution in the strip




In order to prove p:Strip write Ψ(z)=a_Ψ,∞(0,y/ω)+Ψ^⊥(z) with Ψ^⊥ the projection of Ψ to the space orthogonal to the constant functions. The main term will come from the constant term, and we will bound the remaining integral of Ψ^⊥. 
To do this we prove the following lemma.

Let  be conjugate to _0(p) with a cusp at ∞ of width ω.
For any α>7+12θ/3+6θ and any η_1<1/2+4θ, there are norms _α,_α+1 of orders α and α+1 respectively, so that for any Ψ∈ C_c^∞(), for any C≥ 1,  β∈(0,1-η_13) and any 1≤ k<CK_0 we have

    |∫_k^k+1Ψ^⊥(y+iy/T)dy/y|≪ C^1+η_1/2(_α(Ψ)ω^1-η_1-β/k^3-η_1/2T^η_1/2+_α+1ω^3/4-η_1-β/k^5/4-η_1/2 T^1/4+η_1/2).

Here

    K_0=K_0(Ψ,T)=(_∞,1(Ψ)^2_α(Ψ)^2ω^-2βT^η_1)^1/η_1+1.



Fix a large parameter N∈ to be determined later and write 

    ∫_k^k+1Ψ^⊥(y+iy/T)dy/y=∑_j=0^N-1∫_t_j^t_j+1Ψ^⊥(y+iy/T)dy/y,

with t_j=Nk+jN. For t_j≤ y≤ t_j+1 we can approximate 

    Ψ^⊥(y+iy/T)=Ψ^⊥(y+it_j/T)+O(_∞,1(Ψ)/Nk+j),

where we used that, for the hyperbolic distance, 

    d(y+iy/T,y+it_j/T)=|log(y/t_j)|≤log(t_j+1/t_j)≤ (Nk+j)^-1.


It is here that we correct the error in the proof of <cit.>.


Plugging this in gives 

    ∫_t_j^t_j+1Ψ^⊥(y+iy/T)dy/y=∫_t_j^t_j+1Ψ^⊥(y+it_j/T)dy/y+O(_∞,1(Ψ)/(Nk+j)^2).

Now for the first term, expanding Ψ^⊥ into its Fourier series, using integration by parts to bound |∫_t_j^t_j+1e^2π i my/ωdy/y|≪ω/mk, and using l:AveFourier with η_1 and α_1=2α-3 we can bound


    |∫_t_j^t_j+1Ψ^⊥(y+it_j/T)dy/y|   =   |∑_m≠ 0a_Ψ,∞(m,t_j/ω T)∫_t_j^t_j+1e^2π i my/ωdy/y|
       ≪   ω/k∑_m≠ 0|a_Ψ,∞(m,t_j/ω T)|/m
       ≪   ω/k (_α_1(Ψ)(kω T)^η_1+_α_1+1(Ψ)(kω T)^η_1+1/4)
       ≪   ω^1-η_1/k^1-η_1_α_1(Ψ)/T^η_1+ω^3/4-η_1/k^3/4-η_1_α_1+1(Ψ)/T^η_1+1/4.

Summing over 0≤ j<N we get that 

    |∫_k^k+1Ψ^⊥(y+iy/T)dy/y|≪ Nω^1-η_1/k^1-η_1_α_1(Ψ)/T^η_1+Nω^3/4-η_1/k^3/4-η_1_α_1+1(Ψ)/T^η_1+1/4+_∞,1(Ψ)/Nk^2.


First, for k≤ K_0 we can take

    N=N_0(k)=[√(_∞,1(Ψ)_α_1(Ψ))T^η_1/2ω^β k^1+η_1/2]≥ 1,

and defining the norms 

    _α(Ψ)=√(_α_1(Ψ)_∞,1(Ψ)),  _α+1(Ψ)=√(_∞,1(Ψ)_α_1+1(Ψ)^2/_α_1(Ψ)).

(that are indeed of degrees α and α+1), we get that 

    |∫_k^k+1Ψ^⊥(y+iy/T)dy/y|≪_α(Ψ)ω^1-η_1-β/k^3/2-η_1/2T^η_1/2+_α+1(Ψ)ω^3/4-η_1-β/k^5/4-η_1/2 T^1/4+η_1/2.



Next for K_0≤ k<CK_0 we take instead   N=[C^1+η_1/2N_0]≥ 1 to get that 

    |∫_k^k+1Ψ^⊥(y+iy/T)dy/y|≪ C^1+η_1/2_α(Ψ)ω^1-η_1-β/k^3/2-η_1/2T^η_1/2+C^1+η_1/2_α+1(Ψ)ω^3/4-η_1-β/k^5/4-η_1/2 T^1/4+η_1/2.



Using this estimate it is possible to evaluate ∫_1^K_0Ψ^⊥(y+iy/T)dy/y. We now show how to boot strap this to extend the range all the way up to K_1=K_0^1+η/2η.

With the same assumptions and notations as in l:unit, let K_1=K_1(Ψ,T)=K_0(Ψ,T)^1+η_1/2η_1. Then there is a constant c∈ (1/e,1) such that 

    ∫_1^cK_1Ψ(y+iy/T)dy/y≪loglog(K_0)(_α(Ψ)ω^1-η_1-β/T^η_1/2+_α_2(Ψ) ω^3/4-η_1-β-β/4η_1/T^η_1/2+1/8),

where _α is as above and _α_2 is a norm of degree α_2=4η+3/4η+α4η-1/4η.


As a first step, a simple application of  l:unit with C=1, noting that for both terms the power of k in the denominator is greater than one gives

    ∫_1^K_0Ψ(y+iy/T)dy/y   =   ∑_k=1^K_0-1∫_k^k+1Ψ(y+iy/T)dy/y
       ≪   _α(Ψ)ω^1-η_1-β/T^η_1/2+_α+1(Ψ)ω^3/4-η_1-β/ T^1/4+η_1/2.


Next let s_ℓ denote the partial sums of geometric series s_ℓ=∑_j=0^ℓ (1-η_11+η_1)^j converging to s_∞=1+η_1/2η_1 and let C_ℓ=K_0^s_ℓ-1. Applying  l:unit with C=C_ℓ we get the bound

    ∫_C_ℓ-1K_0^C_ℓK_0Ψ(y+iy/T)dy/y   =   ∑_k=C_ℓ-1K_0^C_ℓ K_0-1|∫_k^k+1Ψ(y+iy/T)dy/y|
       ≪   C_ℓ^1+η_1/2_α(Ψ)/(C_ℓ-1K_0)^1-η_1/2ω^1-η_1-β/T^η_1/2+C_ℓ^1+η_1/2_α+1(Ψ)/(C_ℓ-1K_0)^1-2η/4ω^3/4-η_1-β/ T^1/4+η_1/2.

From our choice of the constants we have that C_ℓ=[K_0 C_ℓ-1]^1-η/1+η so that C_ℓ^1+η_1/(C_ℓ-1K_0)^1-η_1=1 and
C_ℓ^1+η_1/(C_ℓ-1K_0)^1-2η/2=(C_ℓ-1K_0)^1/2, hence

    ∫_C_ℓ-1K_0^C_ℓK_0Ψ(y+iy/T)dy/y≪_α(Ψ)ω^1-η_1-β/T^η_1/2+[C_ℓ-1K_0]^1/4_α+1(Ψ)ω^3/4-η_1-β/ T^1/4+η_1/2.

Now bounding C_ℓ-1K_0≤ K_0^1+η_1/2η_1=K_1 and plugging in 
K_0=(_∞,1(Ψ)^2_α(Ψ)^2ω^-2βT^η_1)^1/η_1+1 we can bound the second term by

    K_0^1+η_1/8η_1_α+1(Ψ)ω^3/4-η_1/ T^1/4+η_1/2=
    _α_2(Ψ)ω^3/4-η_1-β-β/4η_1/ T^1/8+η_1/2,

with 

    _α_2(Ψ)=_α+1(Ψ)_∞,1(Ψ)^1/4η_1_α(Ψ)^-1/4η_1.

We thus get that for each ℓ≥ 1 we have

    ∫_C_ℓ-1K_0^C_ℓK_0Ψ(y+iy/T)dy/y≪_α(Ψ)ω^1-η_1-β/T^η_1/2+_α_2(Ψ)ω^3/4-η_1-β-β/4η_1/T^1/8+η_1/2.


Finally, taking ℓ= [loglog(K_0)/log(1+η1-η)]+1 we get that 

    s_ℓ≥1+η_12η_1- (1-η_11+η_1)^ℓ≥1+η_12η_1-1log(K_0),
 
so that 
e^-1 K_1≤ C_ℓ K_0≤ K_1, hence, C_ℓ K_0=cK_1 for some c>1/e.
Summing up these ℓ=O(loglog(K_0)) terms concludes the proof.




For large values of k>K_1 this estimate is no longer optimal, and instead we will use the following alternative bound.

Let  is conjugate of _0(p) with a cusp at ∞ of width ω. For any α_1>5+6θ/3+6θ and η_1<1/2+4θ there are norms _α_1,_α_1+1  such that for any 
Ψ∈ C^∞_c(), for any k≥ 1 we have

    ∫_ω k^ω(k+1)Ψ^⊥(y+iy/T)dy/y≪_α_1(Ψ)/k^2-η_1T^η_1+_α_1+1(Ψ)/k^7/4-η_1T^η_1+1/4+_∞,1(Ψ)/k^2.



For kω≤ y≤ (k+1)ω we can estimate 

    Ψ(y+iy/T)=Ψ(y+ikω/T)+O(_∞,1(Ψ)/k)
    ,
 
so that 

    ∫_ω k^ω(k+1)Ψ^⊥(y+iy/T)dy/y=∫_ω k^ω(k+1)Ψ^⊥(y+iω k/T)dy/y+O(_∞,1(Ψ)/k^2).

Expanding the first term in Fourier series 

    ∫_ω k^ω(k+1)Ψ^⊥(y+iω k/T)dy/y=∑_m≠ 0a_Ψ(m,k/T)∫_ω k^ω(k+1)e^2π i my/ωdy/y.

Integrating by parts we can bound 

    |∫_ω k^ω(k+1)e^2π i my/ωdy/y|=|∫_k^k+1e^2π i mydy/y|
    
    ≤1/mk^2,

and using l:AveFourier we bound

    ∫_ω k^ω(k+1)Ψ^⊥(y+iω k/T)dy/y   =   ∑_m≠ 0a_Ψ(m,k/T)∫_ω k^ω(k+1)e^2π i my/ωdy/y
       ≤   1/k^2∑_m≠ 0|a_Ψ(m,k/T)|/m
       ≪   _α_1(Ψ)/k^2-η_1T^η_1+_α_1+1(Ψ)/k^7/4-η_1T^η_1+1/4.

concluding the proof.


 We can now give the 
 
Noting that  

    ∫_T/√(T^2+1)^∞Ψ(y+iyT)dy/y=∫_1^∞Ψ(y+iy/T)dy/y+O(_∞,0(Ψ)/T^2),

and that 

    ∫_1^∞ a_Ψ,∞(0,y/ω T)dy/y=1/ω∫_0^ω∫_1/T^∞Ψ(x+iy)dydx/y,

we just need to bound ∫_1^∞Ψ^⊥(y+iy/T)dy/y. 

Let β=12-η_1 and let K_0=K_0(Ψ,T)=(_∞,1(Ψ)^2_α(Ψ)^2ω^-2βT^η_1)^1/η_1+1 be as in l:unit and 
K_1=K_0^1+η_1/2η_1=(_∞,1(Ψ)_α(Ψ))^1/η_1ω^-β/η_1T^1/2 be as in p:bootstrap. Then

    ∫_1^cK_1Ψ^⊥(y+iy/T)dy/y≪loglog(K_0)(_α(Ψ)ω^1/2/T^η_1/2+_α_2(Ψ)ω^1/4-1-2η/8η/T^η_1/2+1/8).

Next, 


using Lemma l:period for Ψ̃=Ψ^a_ω having period one, we can bound

    ∫_cK_1^∞Ψ^⊥(y+iy/T)dy/y   =   ∫_cK_1/ω^∞Ψ̃^⊥(y+iy/T)dy/y
       =   ∑_k>cK_1/ω∫_k^k+1Ψ̃^⊥(y+iy/T)dy/y
       ≪   _α_1(Ψ)/T^η_1+_α_1+1(Ψ)/T^η_1+1/4+ω_∞,1(Ψ)/K_1.

The first two terms are clearly bounded by the similar terms appearing above and plugging in the value of K_1, the third term is 

    ω_∞,1(Ψ)/K_1= _∞,1(Ψ)^1-1/η_1_α(Ψ)^1/η_1(ω^η_1+β/T^η_1/2)^1/η_1.

Notice that _∞,1(Ψ)^1-1/η_1_α(Ψ)^1/η_1 is a norm of degree α/η_1-3(1/η_1-1) (which is smaller than α as long as α<3),
and with our choice of β=12-η we see that 

    (ω^η_1+β/T^η_1/2)^1/η_1=(ω^1/2/T^η_1/2)^1/η_1≤ω^1/2/T^η_1/2.

Hence, after perhaps replacing _α by a different norm of degree α we get that 

    ω_∞,1(Ψ)/K_1≤_α(Ψ)ω^1/2/T^η_1/2,

whence

    ∫_1^∞Ψ^⊥(y+iy/T)dy/y≪loglog(K_0)(_α(Ψ)ω^1/2/T^η_1/2+_α_2(Ψ)ω^1/4/T^η_1/2+1/8).

Finally, replacing the exponent η_1 with a slightly smaller exponent we may remove the loglog(K_0) term.






























 
 




 §.§ Two sided cuspidal geodesics

For the cases of interest here, the lattice  has a cusps at ∞ and at 0 and it is natural to consider shears of the two sided cuspidal geodesic connecting them,
that is,

    μ_T(Ψ)=∫_0^∞Ψ(Tx+i y)dy/y.

It is easy to see that μ_T is invariant under scaling, μ_T(Ψ)=μ_T(Ψ^a_t) and a simple computation shows that under 
σ=([  0  1; -1  0 ]) it transforms via μ_T(Ψ^σ)=μ_-T(Ψ). In fact, using that Ψ^σ(z)=Ψ(-1z) and making a change of variables gives the identity.

    μ_T(Ψ)=∫_1√(T^2+1)^∞Ψ(Ty+iy)dy/y+∫_1√(T^2+1)^∞Ψ^σ(-Ty
    +i y)dy/y

Applying our results on shears of cuspidal geodesic rays we get the following. 

Let  be conjugate to _0(p) with cusps at ∞ and 0. Let ω_1,ω_2 denote the widths of the cusps of  and ^σ at ∞ respectively and let ω=√(ω_1ω_2).
For any α>7+12θ/3+6θ, η_1<1/2+4θ and T≥ω^1/η_1 we have 
for any Ψ∈ C^∞_c(), 

    μ_T(Ψ)   =   2μ_(Ψ)log(Tω)+⟨_,∞,Ψ⟩+⟨_,0,Ψ⟩
          +O_α,η_1(_α(Ψ)ω^1/2/T^η_1/2+_α_2(Ψ)ω^1/4/T^η_1/2+1/8),

where _α,_α+1 are as in t:Equidistribution1. 

Note that if Ψ(z) has period ω_1 and Ψ^σ has period ω_2 then Ψ̃(z)=Ψ(√(ω_2ω_1)z) satisfies that both Ψ̃ and Ψ̃^σ have a period of ω=√(ω_1ω_2). Since μ_T(Ψ)=μ_T(Ψ̃), using t:Equidistribution1 with Ψ̃ and Ψ̃^σ in each part of  (<ref>) gives the claimed result.

























































§ LATTICE POINTS IN CONES

We now apply our results on equidistribution of shears to get effective counting estimates for counting lattice points in cones of the form

    _T={z∈: |(z)|≤ T(z)}
    ,
            
    T→∞.

For  conjugate to _0(p) and τ∈_2(), we define the counting function


    __T^τ()=#{∈: τ^-1 i∈_T}.



For any η<3/40+72θ and β_1>1+2θ, we have for T>(_τ_τ)^ that

    __T^τ()=2T/v_(log(T^2_τ_τσ)-2+v_  (_,_τ(i)+_,_τ(i))+O((_τ_τσ)^β_1η/T^η)).

Here _τ denotes the width of the cusp at ∞ of ^τ=ττ^-1, _τ=τ∞, and _τ=τ0. 


As a first step we show that the cones _T are well rounded. 


Let g∈ G such that g i∈_T with T≥ 1. Then for any h∈ B_δ we have gh i∈_T(1+4δ). 


For any h∈ B_δ we have that |(h i)|≤δ and |(h i-1)|≤ 2δ Write g i=x+iy then we can write 
gh i=x+y(ξ+iη)=x'+iy' with |ξ|≤δ and |η-1|≤ 2δ. We can thus write

    |x'|/y'=|x+ξ y|/η y≤|x|/y(1+|ξ y/x|/η).


We now consider two cases, first assume that |x|/y≥ 1, in which case 

    |x'|/y'≤ T(1+δ/1-2δ)≤ T(1+4δ).

Next, when |x|≤ y we bound

    |x'|/y'=|x+ξ y|/η y≤y(1+|ξ|)|/η y≤1+δ/1-2δ≤ 1+4δ.



Now let χ__T denote the indicator function of _T and consider the function 

    F_T,τ(g)=∑_∈χ__T(τ^-1 g.i)

Note that F_T,τ∈ L^2( G/K) and that evaluating at the identity we have F_T,τ(1)=__T^τ().
From the well roundedness of _T we get the following:


For δ>0 small, let ψ_δ∈ C^∞_c(K G/K) be supported on B_δ with ∫_G ψ_δ=1, and let 

Ψ_δ(g)=∑_∈ ψ_(g).
Then Ψ_δ∈ L^2( G/K) and 

    ⟨ F_T(1-4δ),τ,Ψ_δ⟩≤__T^τ()≤⟨ F_T(1+4δ),τ,Ψ_δ⟩,

where the inner product the standard inner product in L^2( G/K).


Unfolding Ψ_δ we can write

    ⟨ F_T,τ,Ψ_δ⟩   =   ∫_G F_T,τ(g)ψ_δ(g)dg.

By l:round we have that for any g∈ B_δ

    F_T(1-4δ),τ(g)≤__T^τ()≤ F_T(1+4δ),τ(g),

concluding the proof.



As is well known, this inner product F_T,Ψ_ not only is an approximation to the sharp cutoff __T(), but it is also itself a smooth counting function, since

    F_T,Ψ_
    
    =
    ∑_∈χ__T(g),

where

    χ__T(g)
    =
    ∫_g∈ Gχ__T( g)ψ_(g)dg,

is a smoothed cutoff, cf. (<ref>).



We thus need to evaluate the inner product  ⟨ F_T,τ,Ψ_δ⟩. The following simple  lemma, relates these to the shears of two sided cuspidal geodesics.

For any Ψ∈ C^∞_c() we have 

    ⟨ F_T,τ,Ψ⟩= ∫_-T^T∫_0^∞Ψ^τ(xy+iy)dyydx



Unfolding F_T,τ and making some changes of variables gives 

    ⟨ F_T,τ,Ψ⟩   =   ∫_G χ__T(τ^-1 gi)Ψ(gi)dg
       =   ∫_G χ__T( gi) Ψ^τ (gi)dg
       =   ∫_0^∞∫_-yT^yTΨ^τ(x+iy)dxdyy^2
       =   ∫_0^∞∫_-T^TΨ^τ (xy+iy)dxdyy
       =   ∫_-T^T∫_0^∞Ψ^τ(xy+iy)dyydx.

as claimed.







































We are in position now to prove thm:2, which follows easily from the following:

Let >0 be sufficiently small and Ψ=Ψ_ as in (<ref>). 
Assume that α>7+12θ/3+6θ and η_1<1/2+4θ.
Then

    F_T,τ,Ψ=
    2T v_(
    log( T^2 _τ_τ)
    -2
    +v_
    (K_^τ,∞,Ψ^τ+K_^τ,0,Ψ^τ)
    )


    1in
    +
    O_,η_1(
    _α(Ψ)(_τ_τσ)^1/4T^1-η_1/2+_α_2(Ψ)(_τ_τσ)^1/8T^7/8-η_1/2)


1in


+
O_η_1(



(ω_τω_τσ)^1/2η_1
log(ω_τω_τσ)
Ψ_∞)
.












By  l:Inner2shear, we have 

    ⟨ F_T,τ,Ψ⟩= ∫_-T^T∫_0^∞Ψ^τ(xy+iy)dyydx
    .


 
Let 

    M=(ω_τω_τσ)^1/2η_1
    .

Then for |x|>M,
we may apply  c:twosided 


to 
 the inner integral to obtain

∫_0^∞Ψ^τ(xy+iy)dyy
   =   
log(|x|^2_τ_τσ)/v_+⟨_^τ,∞,Ψ^τ⟩+⟨_^τ,0,Ψ^τ⟩


      
+O_α,η(_α(Ψ)(_τ_τσ)^1/4/|x|^η_1/2+_α_2(Ψ)(_τ_τσ)^1/8/|x|^1/8+η_1/2)



.
Integrating over M≤ |x|≤ T gives the first three terms in (<ref>).

For |x|<M, we argue as follows. 
First fix x and apply (<ref>) to the  y integral: 

    ∫_0^∞Ψ^τ(xy+iy)dyy
    =
    ∫_1√(1+x^2)^∞Ψ^τ(xy+iy)dyy
    +
    sim
    
    
    
    
    
    
    
    
    
    
    ,

where “sim” refers to a similar term with τ replaced by τ and x by -x.
Break the y integral into 

    ∫_1√(1+x^2)^∞=∫_1/√(x^2+1)^1/_τ+∫_1/_τ^∞.

(If 1/_τ<1/√(x^2+1), then we only keep the second term.)
We claim that the curve in this second term passes through only one “standard” fundamental domain for ^τ (that is, a Dirichlet domain centered at a point in the injective horoball centered at ∞). Indeed,
recall that  is conjugate to _0(p), which is a subgroup of _2(). But the length 1 horocycle centered at ∞ is injective for the latter, and  hence also for the former. The width of the cusp at ∞ of ^τ is _τ, and hence this  length 1 horocycle is at (Euclidean) height y=1/_τ.

On this one fundamental domain, 
 Ψ^τ is supported on a small δ-neighborhood of τ^-1i, which occurs in some range, say, 
 
    y∈(y_0,y_0(1+C)).
 
 
 Hence
 
    ∫_1/_τ^∞Ψ^τ(xy+iy)dyy ≪Ψ_∞∫_y_0^y_0(1+C)dy/y ≪Ψ_∞
    .

 
 For the other interval, we trivially bound

    ∫_1/√(x^2+1)^1/_τΨ^τ(xy+iy)dyy ≪Ψ_∞log(3+x)_τ
    .


Finally, we integrate over |x|<M; since <1, the term MΨ_∞ is not dominant and may be dropped. Repeating the argument with τ replaced by τ gives
 the final term of  (<ref>).





We are finally in position to give the following

Let  and η_1 be as in prop:smoothC. Let η=η_1/2(α+1) and β_1=1/2η_1 and denote by ω=√(ω_τω_τσ).
We may choose our δ-bump function ψ_δ so that 

    _α(Ψ_δ)≍δ^-α,  _α_2(Ψ_δ)≍δ^-α_2,  Ψ_∞≍δ^-2
 
where _α,_α_2 are as in  t:Equidistribution1.

Now we further simplify the  Eisenstein terms appearing in prop:smoothC. Since E_^τ,∞(z,s)=E^τ_,_τ(z,s) removing the residue we get that 
_^τ,∞(z)= ^τ_,_τ(z) and  hence, ⟨_^τ,∞,Ψ_δ^τ⟩_^τ=⟨_,_τ,Ψ_δ⟩_.






Since _,_τ(z) is smooth and Ψ_δ is supported on a δ-neighborhood of i  we can estimate
⟨_,_τ,Ψ_δ⟩_=_,_τ(i)+O(δ) so the Eisenstein term can be approximated by

    ⟨_^τ,∞,Ψ_δ^τ⟩_^τ= _,_τ(i)+O(δ)
    .


Making an optimal choice of 

    δ=ω^1/2(1+α)T^-η_1/2(1+α),

the first error term dominates and we get 

    ⟨ F_T,τ,Ψ_δ⟩   =   2T/v_(log(T^2^2)-2+v_  (_,_τ(i)+_,_τσ(i)))
          +O(ω^2β_1ηT^1-η).



Finally, using (<ref>) relating the counting problem to the inner product 
(after replacing T with T(1± 4δ)) we get that

    __T^τ()=2T/v_(log(T^2)-2+v_  (_,_τ(i)+_,_τ(i))+O((ω^2β_1T)^η).



As can be expected, the main term does not depend on τ. The secondary term does depend on τ, but only involves
knowledge of which cusps are used for the Eisenstein term, and
 the widths of these cusps.
 





§ COUNTING INTEGER SOLUTIONS

In this section, we establish the results claimed in sec:1.3, handling thm:3 and <ref> simultaneously.


 §.§ Decomposition into orbits

Consider the variety  

    V_d :  
     b^2-4ac=d,
 
 where d is fixed.
Identifying the triple (a,b,c) with the quadratic form 

    Q(x,y)=ax^2+bxy+cy^2,
 
gives a natural _2 action on V_d, via

    Q^g(v)=Q(vg^t),
 
where g∈_2 acts linearly on v=(x,y) from the left (here g^t is the transpose of g).
More explicitly, the action of g∈_2 on triples (a,b,c)∈ V_d  is given by the linear action

    (a,b,c)^g:=(a,b,c)ι(g).

where ι:_2→_b^2-4ac is the spin morphism given by

    ι([ a b; c d ])=[   a^2   2ab   b^2;    ac ad+bc    bd;   c^2   2cd   d^2 ].


In particular, _1=_2() acts on the integer points V_d() and we can decompose

    V_d()=∪_i=1^h(d)v_i _1,

into finitely many orbits.
The number h(d) of orbits is, in general, 
 
 very mysterious;
 for instance, when d is a square free fundamental discriminant then h(d) is the class number of the quadratic extension (√(d)). However when d=n^2 is a perfect square, the number of orbits can be computed explicitly. In the following lemma, we compute it and give a full set of representatives for the orbits. 


For d=n^2 a square we have h(n^2)=n. Moreover,
the set {(0,n,0)^τ_j|0≤ j<n} with τ_j=([   1 j/n;   0   1 ]) 
is a full set of representatives for the classes of  V_d()/_2(), 


We identify a point (a,b,c)∈ V_d() with the corresponding quadratic form Q(x,y)=ax^2+bxy+cy^2 having discriminant d=b^2-4ac. Then SL_2() acts on the set of quadratic forms by Q^γ(v)=Q(vγ^t) where v=(x,y) and γ∈_2() is acting linearly on the left.

Recall that a binary quadratic form Q has a square discriminant, if and only if the form factors as a product of linear forms 

    Q(x,y)=(A x+B y)(C x+Dy),
 
in which case the discriminant is given by (AC-BD)^2. We thus get a map, from the set 
_n of 2× 2 matrices with determinant n onto the set of quadratic forms of discriminant n^2, sending M=[ A B; C D ] to the form Q_M(x,y)=(Ax+By)(Cx+Dy) (this map is not injective since the same form can have several factorizations). A direct computation shows that Q_M^γ=Q_Mγ, and since 

    Δ_n={[ A 0; C D ]: AD=n,  0≤ C<D},
 is a full set of representatives for _n/, the set {Q_M, M∈Δ_n} is a full set of representatives for classes of quadratic form of discriminant n^2 (some of these might be equivalent though).
A form in this set of representatives can be written explicitly as

    Q_M(x,y)=Ax(Cx+Dy)=ACx^2+nxy,
 and we see that the classes with A≠ 1 are redundant. A full set of inequivalent representatives is thus given by the forms Cx^2+nxy with 0≤ C<n.  Finally, observing that Cx^2+nxy is equivalent to nxy-Cy^2 concludes the proof.


Instead of looking at all integer points in V_d() one can consider only primitive points (i.e., points with gcd  (a,b,c)=1). It is easy to see that _1 also acts on the set of primitive points, and the same proof shows that the set {(0,n,j)|0≤ c<n,  (n,j)=1} is a full set of representatives for the orbits of primitive points. 


Using our orbit decomposition of V_d we can also get a corresponding decomposition of the variety 

    W_d  :  
    x^2+y^2-z^2=d 
    .

The map  (x,y,z)↦ (z+y/2,x,z-y2) is a bijection between W_d and V_d and the integer points W_d() map to the set 

    V_d():={(a,b,c)∈ V_d| b∈,  a,c∈12,  a+c∈}.

From this map we see that the congruence subgroup 

    _2={∈: γ̅∈{([ 1 0; 0 1 ]),  ([  0  1   ; -1  0 ])}},

(with ∈_2(/2) the projection of ),
acts on V_d() (and hence also on W_d()). Using the classification of the orbit of the _1 action on V_d() we get the following classification for the _2 action on V_d().

For d=n^2 a complete set of representatives for the _2 orbits of V_d() are given by 

    {(0,n,0)^τ_j,  0≤ j<2n}∪{(0,n,0)^τ_j: 0≤ j<2n},

where τ_j=([   1 j/n;   0   1 ]) is as above and τ_j=([ 1+j/2n   j/2n       ;      1      1 ]).


Let (a,b,c)∈V_d() then (2a,2b,2c)∈ V_4d(). From the classification of _1 orbits of V_4d() there is ∈_1 with (a,b,c)=(0,n,j/2)^ with 0≤ j<2n. We can write  as σ with ∈_2 and σ∈{([ 1 0; 0 1 ]), ([ 1 1; 0 1 ]), ([ 1 0; 1 1 ])} in the set of representatives for _1/_2. We can thus write the point (a,b,c) as
(0,n,j/2)^ or as (0,n,n+j/2)^ or (n+j2,n+j,j2)^ for some 0≤ j<2n. Now, (a,b,c)∈V_d() and   preserves this space, so in the first two cases we must have that j/2∈. Hence, a full set of representatives are indeed (0,n,j)=(0,n,0)^τ_j and (n+j2,n+j,j2)=(0,n,0)^σ_j for 0≤ j<2n.


Let B_T={(a,b,c)∈^3: 2a^2+b^2+2c^2<T_2} and note that V_d()∩ B_T is in bijection with the set {(x,y,z)∈ W_d(): x^2+y^2+z^2≤ T^2} so 
_d(T)=#(V_d()∩ B_T). 
Moreover, under this bijection, the primitive points of W_d() correspond exactly to the _2-orbits of the classes (0,n,0)^τ_j and  (0,n,0)^τ_j with (j,n)=1.


Decomposing the integer points of V_d()∩ B_T and V_d()∩ B_T into the finitely many orbits, it is enough to count points in each orbit separately. For this we need to
estimate terms of the form

    #{∈:  (0,n,0)^τ≤ T}

with τ=τ_j or τ=τ_j as above and the lattice  =_1 or =_2 respectively.  
We now show that these counting functions are  given in therms of the cone counting function defined in (<ref>).

For any lattice  and τ∈ G we have

    #{∈: (0,n,0)^τ≤ T}=__T_n^τ^-1()

with T_n=√(T^2/2n^2-1/2).
 

Write τγ=a_yn_xk so that τ i=yx+iy.
Since (0,n,0)^a_y=(0,n,0) and our norm is K-invariant we can explicitly compute 

    (0,n,0)^τ^2=n^2(1+2x^2),
 
so that indeed (0,n,0)^τ≤ T if and only if |x|≤ T_n which is equivalent to τ.i∈_T_n.




 §.§ Square discriminants

Our goal here is to prove thm:4 by estimating

    #V_d()∩ B_T   =   ∑_j=0^n-1#{∈_1:  (0,n,0)^τ_j≤ T}
       =   ∑_j=0^n-1__T_n^τ^-1_j(_1)
    .

Note that _1^τ^-1_j has a cusp at ∞ of width 1 and _1^τ^-1_jσ has a cusp at ∞ of width ω_j=n^2/(n,j)^2.















Hence, from  p:ConeCount we get that

    __T_n^τ_j^-1(_1)=2T_n/v__1( 2log(T_n)+log(n^2/(n,j)^2)+2v___1(i)+O(n^2β_1η/T_n^η)),

with β_1=β-12>1+2θ. Recalling the assumption T≥ d=n^2 in the statement of thm:1, we have that T_n>1 and we can
estimate T_n=T/√(2)n+O(n/T), and 

    log(T_n)=log(T)-12log(2)-log(n)+O(n^2/T^2),
 
so that 

    __T_n^τ_j^-1(_1)=√(2)T/nv__1(2log(T)-log(2)-2log((n,j))+2v___1(i)+O(n^(2β_1+1)ηT^η)).

Summing over all orbits we get that 






    #V_d()∩ B_T=√(72)T/π(log(T)-12log(2)+π 3  __1(i)-1/n∑_j=1^nlog(n,j)+O(n^2βηT^η)).

Plugging in the value of  __1(i) from (<ref>) and noting that 

    ∑_j=1^n-1log(j,n)=∑_a|nϕ(na)log(a),

concludes the proof of (<ref>).



 §.§ Sum of squares

Next we prove thm:3 by estimating _d(T)=V_d()∩ B_T. Again, split the integral points into the finitely many _2 orbit and count in each orbit.
We thus need to estimate __T_n^τ_j^-1(_2) and __T_n^τ_j^-1(_2).



Let  _j=τ_j^-1∞,_j=τ_j^-10 and let  ω_j,ω_j' denote the width of the cusps at ∞ of _2^τ_j^-1 and _2^τ_j^-1σ, appearing in the formula for __T_n^τ_j^-1(_2). Similarly let  _j=
τ_j^-1∞,_j=τ_j^-10 and let  ω_j,ω_j' the corresponding cusp widths. 

Recall that  that _2 has only two inequivalent cusps, one at ∞ and another at 1. After verifying which pair of cusps we get for each orbit, we show that the contribution of the Kronecker terms to the counting function is as follows.

With the above notation we have

    ∑_j=0^2n-1(__2,_j(i)+__2,_j(i)+__2,_j(i)+__2,_j(i))=4n(__2,∞(i)+__2,1(i)).



Since τ_j^-1∞=∞ and τ_j^-1∞=-1 (which is _2 equivalent to the cusp at 1)  we see that _j=∞ and _j=1. 
Next τ_j^-10=jn is _1-equivalent to the cusp at infinity by the action of some =([ a b; c d ])∈_1 such that jn=∞=ab, that is, with a=j(j,n) and b=n(j,n). Such an element  lies in _2 iff nj(j,n)^2 is even, so that 
_j={[             ∞ nj/(n,j)^2=02;             1 nj/(n,j)^2=12;               ].. Similarly, τ_j^-10=-j2n+j is _1-equivalent to the cusp at ∞ by ∈_1 with ab=-j2n+j so that a=j(j,2n) and b=2n+j(j,2n). Hence  can be taken from _2 iff j(j+2n)(j,2n)^2 is even implying that  
_j={[                   ∞ j(j+2n)/(2n,j)^2=02;                   1 (j+2n)j/(2n,j)^2=12;                     ].. We thus get that the term  _j=__2,_j(i)+__2,_j(i)+__2,_j(i)+__2,_j(i) is given by

    _j=
    {[             3__2,∞(i)+__2,1(i) nj/(n,j)^2=j(j+2n)/(2n,j)^2=02;             __2,∞(i)+3__2,1(i) nj/(n,j)^2=j(j+2n)/(2n,j)^2=12;            2__2,∞(i)+2__2,1(i)  nj/(n,j)^2≠j(j+2n)/(2n,j)^22. ].

Writing n=2^am with m odd we see that the first case happens when j=02^a but j≠ 02^a+1 (hence for n/2^a values of 0≤ j<2n), the second case when j=02^a+1 (for another n/2^a values of j) and the last when j≠ 02^a (for 2n(1-1/2^a) values of j). Now summing over all 0≤ j<2n we get our result.






Summing up the contributions from the widths of the cusps, we get

With notation as above, let n=2^ν m with m odd. Then 

    ∑_j=0^2n-1log(ω_jω_j'ω_jω_j')=
    8nlog(2n)-2nlog(2)2^ν-8∑_a|nϕ(na)log(a).




Fix 0≤ j<2n. Since τ_j commutes with N the width of the cusp at ∞ of _2^τ_j^-1 is the same as for _2 so ω_j=2. 
Similarly, τ_j^-1[ 1 1; 0 1 ]τ_j=[  2  1; -1  0 ]∈_2 
so ω_j=1. 

Next ω_j' is the smallest integer k such that 

    τ^-1_jσ[ 1 k; 0 1 ]στ_j=[    1-kjn -j^2kn^2;        k    1+kjn ]∈_2,

hence 

    ω_j'={[  n^2/(n,j)^2,  jn(j,n)^2=12; 2n^2/(n,j)^2, jn(j,n)^2=02. ].




Similarly, ω_j' is the smallest integer k such that 

    τ^-1_jσ[ 1 k; 0 1 ]στ_j=[  1+kj2n(1+j2n)       j^2k4n^2;     k(1+j2n)^2 1+kj2n(1+j2n). ]∈_2,

and a similar computation gives that 

    ω_j'={[        4n^2/(2n,j)^2, j(j,2n)=122n(j,2n)=02;        8n^2/(2n,j)^2,                      ;                       ].


Writing n=2^ν m with m odd, it is not hard to see that

    ω_jω_jω_j'ω_j'={[  8n^4/(n,j)^4,         j=02^ν; 16n^4/(n,j)^4,                ].

so that 

    ∑_j=0^2n-1log(ω_jω_j'ω_jω_j')
       =   ∑_j=0^2n-1log(8n^4/(n,j)^4)+2nlog(2)(1-12^ν)
       =   2nlog(8n^4)-8∑_j=0^n-1log((n,j))+2nlog(2)-2nlog(2)2^ν
       =   8nlog(2n)-2nlog(2)2^ν-8∑_a|nϕ(na)log(a)
    ,

as claimed.





Combining these results we can express the counting function 

For any η<η_0 for any d=n^2 a perfect square, we have 

    _d(T)=√(32)T/π(log(T)+C_2-log(2)/2^ν_2(n)+1-2/n∑_a|nϕ(na)log(a)+O((Tn)^-η))

where ν_2(n) is the 2-adic valuation of n and 

    C_2=4 -4ζ'/ζ(2)-2-log(4|η(i+1)η(i+12)|^4).

 



 
 Partitioning V_d∩ B_T into _2 orbits and summing in each orbit gives 
 
    _d(T)=∑_j=0^2n-1__T_n^τ_j^-1(_2)+__T_n^τ_j^-1(_2).

 Using  p:ConeCount and the estimates T_n=T/√(2)n+O(n^2T^2) and |ω_jω_j'|≤ n^2 we estimate each of the cone counting functions 
 
    __T_n^τ_j^-1(_2)=√(2)T/π n(log(T^2ω_j ω_j'2n^2)-2+ π(__2,_j(i)+__2,_j(i))+O(n^(2β_1+1)ηT^η)),

 with β_1=β-12 as before, 
 and similarly for __T_n^τ_j^-1(_2). Summing over 0≤ j<2n, by  l:Ksum the contribution of the Kronecker terms is 
 √(32)T(__2,∞(i)+__2,1(i)) and by  l:sumwj the contribution of the terms log(ω_jω_j'ω_jω_j') is 
 
    ∑_j=0^2n-1log(ω_j ω_j' ω_j ω_j')=8nlog(2n)-2nlog(2)2^ν-8∑_a|nϕ(na)log(a),

 so that 

    _d(T)   =   √(32)T/π(2log(T)-2+log(2)+π(__2,∞(i)+__2,1(i))
          -log(2)2^ν+1
    -2/n∑_a|nϕ(na)log(a)+
    O_η(n^2βT^η)).


We now want to express the Kronecker terms in terms of special values of Dedekind eta function. To do this, note that _2=_0(2)^τ with τ=([ 1 0; 1 1 ]) and hence E__2,(z,s)=E__0(2),τ(τ z,s) and also __2,(z)=__0(2),τ(z,s). In particular 

    __2,∞(i)+__2,1(i)=__0(2),0(i+12)+__0(2),∞(i+12).

Using  p:KroneckerLimit for =_0(2) we have 

    _,∞(z)=1/π(2 -2 ζ'/ζ(2)-log(4y|η(2z)|^8|η(z)|^4)- 8log(2)/3),

and

    _,0(z)=1/π(2 -2 ζ'/ζ(2)-log(4y|η(z)|^8|η(2z)|^4)+ log(2)/3).

Hence

    π(_,0(i+12)+_,∞(i+12))=4-4 ζ'/ζ(2)-13log(2)3-2log(|η(i+1)η(i+12)|^2).

Using the transformation law for the Dedekind Eta function 

    |η(z+1)|^2=|η(z)|^2,   |η(-1z)|^2=|z||η(z)|^2,

we have that |η(i+1)η(i+12)|^2=√(2)|η(i)|^4=√(2)(1/4)^4/16π^3 so that 

    π(_,0(i+12)+_,∞(i+12))=4-4 ζ'/ζ(2)-13log(2)3-2log(2)-2log((1/4)^4/16π^3).

and plugging this back in we get that 

    _d(T)   =   √(128)T/π(log(T)+C
    -1/n∑_a|nϕ(na)log(a)+log(2)(13-12^ν+2)+
    O(n^2βηT^η))

with the constant 

    C=2-1-2 ζ'/ζ(2)-5log(2)2-log((1/4)^4/16π^3),

as before.
This completes the proof.




 
 
 
 
 
 
 
 
 
 
 
 
 

 


§ EISENSTEIN SERIES FOR _0(P)

As for the full modular group, _1=_2(), the theory of Eisenstein series for the congruence groups _0(p) is also well understood, in particular, the Fourier coefficients can be expressed explicitly and there is an explicit formula for the Kronicker limit. Since we could not found a suitable reference for these formulas we will include
 
 short proofs here, but we claim no originality. 


We first note that when  is a finite index subgroup of _1, one can express the Eisenstein series for _1 in terms of the Eisenstein series corresponding to the different cusps of . Explicitly, we have

Let σ_1,…, σ_k denote a complete set of representatives of _1/, and let _i=σ_j^-1∞ (these are not necessarily inequivalent cusps for ). Then

    E__1(z,s)=∑_j=1^k ω_j^s-1E_,_j(z,s)

where ω_j denotes the width of the cusp _i.


Since N∩σ_jσ_j^-1⊆ N∩σ_j_1σ_j^-1=N∩_1 it is generated by ([   1 ω_j;   0   1 ]) 
where the width ω_j∈ of _j is the index of __j in _1∩ N.
We can thus write for (s)>1

    E__1(z,s)   =   ∑_(_1∩ N)( z)^s
       =   ∑_j=1^k ω_j^-1∑_∈__i(σ_i z)^s

Since τ__i=σ_i^-1a_ω_i is a scaling matrix for _j we have

    ∑_∈__i(σ_i z)^s=∑_∈__i(a_ω_iτ__i^-1 z)^s=ω_i^sE_,_i(z,s),

and the result follows.

Subtracting the residue and taking the limit as s→ 1 we get the following

For  a subgroup of _1 the Kronecker limit satisfies

    __1(z)=∑_j=1^k _,_i(z)+3/π k∑_j=1^klog(ω_j)


In particular, applying this to the subgroup =_0(p) of _1=_2() we get the following identities


    E__1,∞(z,s)=E__0(p),∞(z,s)+p^sE__0(p),0(z,s),


    __1(z)=__0(p),∞(z)+ p__0(p),0(z)+3plog(p)/(p+1)π











 §.§ Fourier Coefficients

For each pair of cusps , the Fourier expansion of the Eisenstein series E_, with respect to the cusp at , is given by

    E^τ__,(z,s)=δ_,y^s+ϕ_,(s)y^1-s+∑_m≠ 0a_,(s;m,y)e(mx),

and since E_,(z,s) is an Eigenfunction with eigenvalue s(1-s) we can write

    a_,(s;m,y)=ϕ_,(s;m)2√(y)K_s-1/2(2π my).


For the full modular group _1=_2() there is just one cusp at ∞ and the Fourier coefficients are given explicitly by
ϕ(s)=ζ^*(2s-1)/ζ^*(2s) and

    ϕ(s,m)=τ_s-1/2(m)/ζ^*(2s),

where ζ^*(s)=π^-s/2ζ(s)(s/2) is the completed Riemann zeta function and  τ_s(m)=∑_ab=|m|( ab)^s is the divisor function 
<cit.>.



For the congruence groups _0(p) the Fourier coefficients can also given by a similar formula and satisfy a similar bound. 


For =_0(p) we have 

    ϕ_∞,∞(s;m)=ϕ_0,0(s;m)=1/p^2s-1{[               -ϕ(s;m)               (p,m)=1; p^s+1/2ϕ(s;mp)-ϕ(m;s)                   p|m ].


    ϕ_∞,0(s;m)=ϕ_0,∞(s;m)=1/p^2s-1{[             -p^sϕ(s;m)                (p,m)=1; p^sϕ(s;m)- √(p)ϕ(s;mp) ].



Since the scaling matrix τ_0 normalizes =_0(p) we have that 
E_,∞^τ_0(z,s)=E_,0(z,s) implying that ϕ_∞,0(s;m)=ϕ_0,∞(s;m), and since τ_0^2=1 is the identity then 
E_,0^τ_0(z,s)=E_,∞(z,s) implying that ϕ_∞,∞(s;m)=ϕ_0,0(s;m).

Now looking at the expansion at infinity of  (<ref>) we get that 

    ϕ(s;m)=ϕ_∞,∞(s;m)+p^sϕ_0,∞(s;m),

and the expansion at 0 gives 

    √(p)ϕ(s;mp)=ϕ_∞,0(s;m)+p^sϕ_0,0(s;m),

where it is understood that ϕ(s;mp)=0 when (p,m)=1. We thus get that

    [   1 p^s; p^s   1 ][ ϕ_0,0(s;m); ϕ_0,∞(s;m) ]=
    [      ϕ(s;m); √(p)ϕ(s;mp) ].

and inverting the matrix concludes the proof.

















 §.§ Kronecker limits

The Kronecker limit corresponding to a cusp  is defined as the limit

    _,(z)=lim_s→ 1(E_,(s,z)-1/v_(s-1)).


When _1=_2() the Kronecker limit formula expresses __1(z) explicitly in terms of the Dedekind η-function
 (see, e.g., <cit.>)

__1(z)
 = 3π
(
2-2'(2)
-log(4y|η(z)|^4)
)
,    where =0.577⋯ is Euler's constant, (s) is the Riemann zeta function, and η(z) is the Dedekind eta function.
In particular, using the special value η(i)^4=(1/4)^4/16π^3 we see that at z=i we have

__1(i)
 = 3π
(
2-2'(2)
-log((1/4)^4/4π^3)
)


One can derive a similar formula for the congruence groups =_0(p). 

For =_0(p) we have 

    _,∞(z)=3/(p+1)π(2 -2 ζ'/ζ(2)-log(4y|η(pz)|^4p/p-1|η(z)|^4/p-1)- 2log(p)p^2/(p^2-1)),

and

    _,0(z)=3/(p+1)π(2 -2 ζ'/ζ(2)-log(4y|η(z)|^4p/p-1|η(pz)|^4/p-1)+ log(p)(p-1)^2/(p^2-1)).




Since representatives for _∞ are given by matrices ([ * *; c d ]) with c≥ 0 integer with c=0p and d∈ with (c,d)=1, after
multiplying the Eisenstein series by 

    ζ_p(2s)=∑_(n,p)=1n^-2s=ζ(2s)(1-p^-2s),
 
and expanding we get for (s)>1

    ζ_p(2s)E_,∞(z,s)   =   ζ_p(2s)y^s+∑_(n,p)=11/n^2s∑_(c,d)=1_c=0py^s/|cz+d|^2s
       =   ζ_p(2s)y^2+∑_c=1^∞_c=0p∑_(d,p)=1y^s/|cz+d|^2s
       =    y^sζ_p(2s)+∑_c=1^∞_c=0 p∑_d∈y^s/|cz+d|^2s- 1/p^2s∑_c=1^∞∑_d∈y^s/|cz+d|^2s
    .

Using Poisson summation on the inner sum gives

    ∑_d∈1/|cz+d|^2s=√(π)(s-12)/(s)(cy)^1-2s+(cy)^1-2s∑_m≠ 0e(-mcx)∫_e(mcyt)/(1+t^2)^sdt

and dividing by ζ_p(2s) we see that 

    E_,∞(z,s)   =   
    y^s+(p-1)√(π)(s-12)ζ(2s-1)(p^2s-1)(s)ζ(2s)y^1-s
          +py^1-s/(p^2s-1)ζ(2s)∑_c=1^∞ c^1-2s∑_m≠ 0e(-mpcx)∫_e(mpcyt)/(1+t^2)^sdt 
          -y^1-s/(p^2s-1)ζ(2s)∑_c=1^∞ c^1-2s∑_m≠ 0e(-mcx)∫_e(mcyt)/(1+t^2)^sdt
    .

The only pole comes from the term containing ζ(2s-1)=1/2(s-1)++O(s-1). Subtracting the residue 
_s=1E_,∞(z,s)=3/π(p+1) and taking the limit as s→ 1 we get that 

    _,∞(z)   =   
    y+lim_s→ 1((p-1)√(π)(s-12)ζ(2s-1)/(p^2s-1)(s)ζ(2s)y^1-s-3/π(p+1)(s-1))
          +p/(p^2-1)ζ(2)∑_c=1^∞1/c∑_m≠ 0e(-mpcx)∫_e(mpcyt)/(1+t^2)dt 
          -1/(p^2-1)ζ(2)∑_c=1^∞1/c∑_m≠ 0e(-mcx)∫_e(mcyt)/(1+t^2)dt
    .

We can evaluate the integral 

    ∫_e(at)/(1+t^2)dt=π e^-2π |a|

to get that 

    ∑_c=1^∞1/c∑_m≠ 0e(-mpcx)∫_e(mpcyt)/(1+t^2)dt    =   π∑_c=1^∞1/c(∑_m=1^∞ e^2π i mpc z+ e^-2π  i mpc z̅)
       =   π∑_m=1^∞∑_c=1^∞1/c(e^2π i mpc z+ e^-2π  i mpc z̅)
       =    -π∑_m=1^∞(log(1-e^2π i mp z)+log(1-e^-2π  i mp z̅))
    .


Recalling the Dedekind η-function is given by

    η(z)=e^π i z/12∏_m=1^∞(1-e^2π i m z),
 
we have that 
∑_m=1^∞log(1-e^2π i m z)=log(η(z))-π i z12  
hence 

    ∑_c=1^∞1/c∑_m≠ 0e(-mpcx)∫_e(mpcyt)/(1+t^2)dt    =   
    
    
     -πlog(|η(pz)|^2)-π^2  p y6
    .

Similarly, we also have 

    ∑_c=1^∞1/c∑_m≠ 0e(-mcx)∫_e(mcyt)/(1+t^2)dt   =   -πlog(|η(z)|^2)-π^2   y6

so that 

    _,∞(z)   =   lim_s→ 1((p-1)√(π)(s-12)ζ(2s-1)/(p^2s-1)(s)ζ(2s)y^1-s-3/π(p+1)(s-1))
          +6log(|η(z)|^2|η(pz)|^-2p)/π(p^2-1)
    .


Next, to compute the limit, write ζ(2s-1)=12(s-1)+γ+O(s-1) to get that the limit above is given by

    6/(p+1)π+ lim_s→ 1(p-1)y^1-s√(π)(s-12)2(p^2s-1)(s)ζ(2s)-3/π(p+1)/s-1   =   6/(p+1)π+d/ds_|_s=1((p-1)y^1-s√(π)(s-12)2(p^2s-1)(s)ζ(2s))
    .

Finally, evaluate the derivative at s=1 

    d/ds_|_s=1((p-1)y^1-s√(π)(s-12)2(p^2s-1)(s)ζ(2s))
    
    
    
    
    
    
       =   3/(p+1)π(log(y)- 2log(p)p^2/(p^2-1)-2log(2)-2 ζ'/ζ(2) )

to get that 

    _,∞(z)   =   3/(p+1)π(2-2 ζ'/ζ(2)-log(4y|η(pz)|^4p/p-1/|η(z)|^4/p-1)- 2log(p)p^2/(p^2-1)).

The formula for the cusp at 0 now follows from (<ref>) and (<ref>).







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