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Theta functions, fourth moments of eigenforms, and the sup-norm problem I

Ilya Khayutin, Raphael S. Steiner

TL;DR

The paper addresses sharp sup-norm bounds for holomorphic Hecke eigenforms in the large-weight limit by deriving sharp fourth-moment estimates through a theta correspondence framework. It replaces amplification with a Bergman–theta kernel construction tied to the Weil representation, enabling a reduction of the fourth-moment bound to a second-moment quaternion count that is proved essentially sharp in both split and division quaternion algebras. The approach yields a decomposed spectral and geometric expansion of the theta kernel, enabling explicit Hecke-equivariant lifts and connections to Jacquet–Langlands transfers, and it extends previous results to co-compact lattices. A notable byproduct is new elementary theta series for signature $(2,2)$ quadratic forms, with potential implications for Lindelöf-on-average phenomena and subconvex-type bounds in the weight aspect. Overall, the work provides a sharp, representation-theoretic route to sup-norm and moment problems in arithmetic hyperbolic geometry, with concrete second-moment matrix counts as the central analytic input.

Abstract

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf{G} \times \mathbf{G} \times \mathbf{SL}_{2}$, for $\mathbf{G}$ an indefinite inner-form of $\mathbf{SL}_2$ over $\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.

Theta functions, fourth moments of eigenforms, and the sup-norm problem I

TL;DR

The paper addresses sharp sup-norm bounds for holomorphic Hecke eigenforms in the large-weight limit by deriving sharp fourth-moment estimates through a theta correspondence framework. It replaces amplification with a Bergman–theta kernel construction tied to the Weil representation, enabling a reduction of the fourth-moment bound to a second-moment quaternion count that is proved essentially sharp in both split and division quaternion algebras. The approach yields a decomposed spectral and geometric expansion of the theta kernel, enabling explicit Hecke-equivariant lifts and connections to Jacquet–Langlands transfers, and it extends previous results to co-compact lattices. A notable byproduct is new elementary theta series for signature quadratic forms, with potential implications for Lindelöf-on-average phenomena and subconvex-type bounds in the weight aspect. Overall, the work provides a sharp, representation-theoretic route to sup-norm and moment problems in arithmetic hyperbolic geometry, with concrete second-moment matrix counts as the central analytic input.

Abstract

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on , for an indefinite inner-form of over , based on the Bergman kernel, and considering its -norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature .

Paper Structure

This paper contains 29 sections, 50 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. General Method
  3. The Weil Representation and Theta Series
  4. Inner-forms of SL2
  5. Normalization of Haar Measures.
  6. Local Weil Representation
  7. The Archimedean Weil Representation
  8. The Non-archimedean Weil Representation
  9. Local Uniformity
  10. Global Weil Representation and Theta Series
  11. Theta Kernels for Eichler Orders
  12. Weil Action on Eichler Orders
  13. Theta Series for Eichler Orders
  14. The Weil L2-norm of the Theta Kernel
  15. The Theta Lift
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let the arithmetic lattice $\Gamma < \mathbf{SL}_{2}(\mathbb{R})$ be the unit norm elements of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$ and $\{f_j\}_j \subset S_m^{\mathrm{new}}(\Gamma)$ be an orthonormalwith respect to the probability measure basis of Hecke newforms of where $\mathrm{ht}_{\Gamma}(z)=1$ if $\Gamma$ is co-compact and if $\Gamma < \mathbf{SL}_2(\mathbb

Theorems & Definitions (126)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 116 more