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Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

Félicien Comtat, Jolanta Marzec-Ballesteros, Abhishek Saha

TL;DR

This work establishes sharp, GRH-conditional bounds for the Fourier coefficients and the global sup-norm of Siegel cusp forms of degree 2 on ${\rm Sp}_4({\mathbb Z})$ that are not Saito–Kurokawa lifts. The authors relate Fourier coefficients to Bessel periods via refined Gan–Gross–Prasad identities, and develop deep local bounds for Bessel functions and integrals using Sugano’s generating functions to handle ramified and non-fundamental discriminants. They combine these local studies with automorphic representation theory and GRH to derive a uniform square-sum bound and, crucially, a pointwise bound on $|a(F,S)|/\|F\|_2$ that yields a global sup-norm bound $\| (\det Y)^{k/2} F \|_\infty \ll_\varepsilon k^{5/4+\varepsilon}$. The results significantly advance non-lift Siegel cusp form bounds by connecting Fourier data to central $L$-values and providing depth-sensitive control of local Bessel factors, with near-cusp improvements in the supremum norm.

Abstract

Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_ε\frac{k^{1/4+ε} (4π)^k}{Γ(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+ε}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_εk^{\frac54+ε}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.

Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

TL;DR

This work establishes sharp, GRH-conditional bounds for the Fourier coefficients and the global sup-norm of Siegel cusp forms of degree 2 on that are not Saito–Kurokawa lifts. The authors relate Fourier coefficients to Bessel periods via refined Gan–Gross–Prasad identities, and develop deep local bounds for Bessel functions and integrals using Sugano’s generating functions to handle ramified and non-fundamental discriminants. They combine these local studies with automorphic representation theory and GRH to derive a uniform square-sum bound and, crucially, a pointwise bound on that yields a global sup-norm bound . The results significantly advance non-lift Siegel cusp form bounds by connecting Fourier data to central -values and providing depth-sensitive control of local Bessel factors, with near-cusp improvements in the supremum norm.

Abstract

Let be an -normalized Siegel cusp form for of weight that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound where denotes the gcd of the entries of , and that its global sup-norm satisfies the bound The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.
Paper Structure (24 sections, 21 theorems, 169 equations, 1 table)

This paper contains 24 sections, 21 theorems, 169 equations, 1 table.

Table of Contents

  1. Introduction
  2. Local calculations
  3. Basic facts and definitions
  4. Preliminaries
  5. The Bessel subgroup
  6. Bessel models
  7. Subgroups and characters of $T({\mathfrak o})$
  8. Main results
  9. Growth of the local Bessel function
  10. Growth of the local Bessel integral
  11. Proof of Proposition \ref{['p:localboundbesselsugano']}
  12. A formula due to Sugano
  13. The required bound
  14. Bounding the local Bessel integral
  15. A set of coset representatives
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $F \in S_k(\Gamma)$ be a Hecke eigenform with Fourier expansion given by e:fouriercoeffsiegelform. Assume that $F$ is not a Saito--Kurokawa lift and let $\pi$ be the automorphic representation generated by $F$. Let $D<0$ be an integer that is congruent to 0 or 1 mod 4 and let $L$ be a positive i

Theorems & Definitions (37)

  • Theorem 1.1: see Theorem \ref{['t:mainfouriergen']}
  • Corollary 1.2
  • Theorem 1.3: see Theorem \ref{['t:mainsup']}
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 27 more