Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) II

Abstract

Given a maximal integral lattice L of signature (m+, 2−) with an odd m 3, we consider the holomorphic cusp forms F of weight l on the bounded symmetric domain of type IV of dimension m with respect to the discriminant subgroup of the orthogonal group O(L ) defined by L . Under a non-negativity assumption on the central L-values, we prove an equidistribution result of Satake parameters in an ensemble constructed from the central values of standard L-functions and the square of the Whittaker-Bessel periods.

Weighted equidistribution theorem for Siegel modular forms of degree 2

“…First we recall the notation and main result from [22] in a special setting. Let Q 5 be the space of column vectors of degree 5 viewed as a quadratic space with the quadratic form t XQY , where…”

“…In the work [12], Kowalski-Saha-Tsimerman investigated the quantity ω Φ l,D,χ from a statistical point of view, including the asymptotic behavior of the average of spinor L-values L f (s, π Φ ) for s on the convergent range of the Euler product taken over the ensemble {ω Φ l,D,χ | Φ ∈ F l } with growing l. Later, the asymptotic formula for the central spinor L-values is proved by Blomer in [4], where even a second moment formula is erabolated by a deep analysis of diagonal and off-diagonal cancellation of terms from the Petersson formula for Siegel modular forms. In our previous paper [22], based on a different technique involving the archimedean Shintani functions and Liu's computation of local Bessel priods for spherical functions, we extend the (first moment) asymptotic formula for central standard L-values of cusp forms on SO(2, m) (m 3) in a general setting. In this paper, we examine the case when m = 3 in detail.…”

“…Let L f (s, AI(χ)) be the Hecke L-function (degree 2) of the automorphic representation AI(χ). By transcribing [22,Theorem 1] in the language of Siegel modular forms, we have the following result.…”

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

“…In the case of our interest, namely holomorphic Siegel modular forms of degree 2, none of the above-mentioned results are known outside of the Maaß space, even though there are some average results [9] (vertical Sato-Tate on average) and [16] (Sato-Tate on average). There are far fewer results however, when one fixes the modular form.…”