Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Abstract

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of Lfunctions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.

Moments of Spinor L-Functions and Symplectic Kloosterman Sums

“…By this local approach, Dickson et al [7] construct an orthogonal basis of S p2q k pN q new,T such that every element is an eigenform of the T 2 ppq operator for all p | N . The latter means locally that at places p | N the adelization is an eigenform of a certain local endomorphism T 1,0 .…”

“…The proof of Böcherer's conjecture in [7] and [8] is obtained via local computations. In the introduction, we already stated relations for newforms; now we present similar results for members of the oldspace basis constructed in the previous section.…”

“…Let S p2q k pN q new,T denote the space of newforms orthogonal to Saito-Kurokawa lifts. For F P S p2q k pN q new,T that satisfy w F,N ‰ 0, we have by [7,Theorem 1.12] that…”

“…For k ě 10 and a prime N " 3 pmod 4q it holds that ÿ F PB p2q k pN q new w F,N Lp1{2, Fˆχ q1 qLp1{2, Fˆχ q2 q " main term`O q1,q2,k pN´α`ǫq, (6) where the main term is the residue at s " t " 0 of the expression (49) and α " 1 2 for k ě 20 and α " k´9 k`1 for k ď 18. In particular, if q 1 " q 2 " 1, the main term equals 4 3 Lp1, χ´4q 2 P 1 plog N q (7) for a certain monic polynomial P 1 of degree 3 depending on k.…”

“…If f is the elliptic modular newform corresponding to the lift F , then w F,N is related to central L-values of f , i.e. by [7,Theorem 3.12] we have that…”