Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects

Abstract

We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms. NotationThe algebraic group GSp 4 is defined as GSp 4 = {g ∈ GL 4 ; t gJg = λ(g)J for some λ(g) ∈ GL 1 }, where J = −1 2 1 2 .

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

“…Averages of L-functions. By combining Theorem 3.10 with some results proved in [24] and [10], we obtain some quantitative asymptotic formulas for averages of L-functions. For example, if B T k is an orthogonal Hecke basis for the space of Siegel cusp forms of full level and weight k > 2 that are not Saito-Kurokawa lifts, then we show that Conjecture 1.12 implies:…”

An Equidistribution Theorem for Holomorphic Siegel Modular Forms for and Its Applications

“…We stated our result only for (N ) for simplicity. In weight aspect we expect that our result holds for other congruence subgroups such as 0 (N ) = [36] (in level one case) and Dickson [18] considered weighted one-level density of spinor L-functions of scalar-valued Siegel cusp forms, namely, let F k (N ) be a basis of the space of Siegel eigen cusp forms of weight k with respect to 0 (N ). Then…”

“…, there exist constants c min (u 1 ) and c reg (u 1 ) (which do not depend on S) such that a G 1 (S, u 1 (1, 0) 18) where χ = v χ v runs over all non-trivial quadratic characters such that χ v is unramified for any v ∈ S, and we set χ S = v∈S χ v (see [29,Example 3.9]). Representative elements of non-trivial unipotent…”

Correction to: On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

“…Secondly, in the statement of Lemma 4.2, the plus sign before the last term in the exponential is to be replaced by the minus sign. These two misprints were pointed out by Dickson [2].…”

“…be a linear combination of f ∈ S k ( (2) 0 (N )) by Poincaré series. Then the exact condition is to be: "for each Q j (1 ≤ j ≤ J ), there is no U ∈ GL(2, Z) satisfying U Q j t U = T ".…”