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An asymptotic formula of spectral average of central $L$-values on ${\bf GSp}(2)$ for square free levels

Seiji Kuga, Masao Tsuzuki

TL;DR

The paper develops a novel relative trace formula for ${f PGSp}_2$ that uses Bessel periods and Rankin–Selberg integrals to study the spectral behavior of central spinor L-values for Siegel cusp forms with square-free level. By translating automorphic data through an exceptional isomorphism to an orthogonal setting, the authors derive an explicit main-term formula with a sharp error term and establish a weighted density theorem for Satake parameters, enabling equidistribution results in the weight and level aspects. A key application proves simultaneous non-vanishing of multiple twisted L-values and yields the existence of infinite families of regular algebraic self-dual ${f GL}_4$ representations with large Hecke fields. The approach blends intricate local orbital integrals, stable integrals, and carefully controlled archimedean and p-adic analysis, culminating in a concrete S-factor evaluation and a robust error analysis. This work advances the automorphic density paradigm for ${f PGSp}_2$ and provides new avenues for non-vanishing results and explicit GL4 constructions with arithmetic significance.

Abstract

We develop a new kind of relative trace formulas on ${\bf PGSp}_2$ involving the Bessel periods and the Rankin-Selberg type integral a la Piatetski-Shapiro for Siegel cusp forms on its spectral side. As an application, a version of weighted equidistribution theorems for the Satake parameters of Siegel cusp forms of square-free level and of scalar weights is proved.

An asymptotic formula of spectral average of central $L$-values on ${\bf GSp}(2)$ for square free levels

TL;DR

The paper develops a novel relative trace formula for that uses Bessel periods and Rankin–Selberg integrals to study the spectral behavior of central spinor L-values for Siegel cusp forms with square-free level. By translating automorphic data through an exceptional isomorphism to an orthogonal setting, the authors derive an explicit main-term formula with a sharp error term and establish a weighted density theorem for Satake parameters, enabling equidistribution results in the weight and level aspects. A key application proves simultaneous non-vanishing of multiple twisted L-values and yields the existence of infinite families of regular algebraic self-dual representations with large Hecke fields. The approach blends intricate local orbital integrals, stable integrals, and carefully controlled archimedean and p-adic analysis, culminating in a concrete S-factor evaluation and a robust error analysis. This work advances the automorphic density paradigm for and provides new avenues for non-vanishing results and explicit GL4 constructions with arithmetic significance.

Abstract

We develop a new kind of relative trace formulas on involving the Bessel periods and the Rankin-Selberg type integral a la Piatetski-Shapiro for Siegel cusp forms on its spectral side. As an application, a version of weighted equidistribution theorems for the Satake parameters of Siegel cusp forms of square-free level and of scalar weights is proved.
Paper Structure (45 sections, 71 theorems, 392 equations)

This paper contains 45 sections, 71 theorems, 392 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of paper
  4. Preliminaries
  5. Distinguished subgroups
  6. Exceptional isomorphism
  7. Height functions
  8. maximal compact subgroups
  9. Haar measures
  10. Hecke algebra
  11. Bessel periods of automorphic forms
  12. Godement section
  13. Section for the archimedean place
  14. Eisenstein series and Rankin-Selberg integrals
  15. Smoothed Rankin-Selberg integral: the spectral side
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $E={\mathbb Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$, and $\Lambda$ a character of ${\rm Cl}(E)$. Let $\mu:{\mathbb A}^\times/{\mathbb Q}^\times{\mathbb R}_{>0} \rightarrow {\mathbb C}^1$ be a character whose conductor $M$ is a product of odd primes inert in $E$; as su uniformly for $l\in 2{\mathbb Z}_{\geqslant 8}$ and all square free positive integer $N$ relatively

Theorems & Definitions (126)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 116 more