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Mixed moments of Hecke eigenforms and $L$-functions

Bingrong Huang

Abstract

In this paper, we establish estimates for the expectation and variance of the mixed $(2,2)$-moment of two Hecke eigenforms of distinct weights. Our results yield applications to triple product $L$-functions. The proofs are based on moments of $L$-functions.

Mixed moments of Hecke eigenforms and $L$-functions

Abstract

In this paper, we establish estimates for the expectation and variance of the mixed -moment of two Hecke eigenforms of distinct weights. Our results yield applications to triple product -functions. The proofs are based on moments of -functions.

Paper Structure

This paper contains 25 sections, 13 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. The variance
  3. An application to the triple product $L$-functions
  4. Moments of $L$-functions
  5. Plan for this paper
  6. Preliminaries
  7. $L$-functions
  8. The Petersson trace formula
  9. Fourier coefficients
  10. An average of the $J$-Bessel function
  11. The variance
  12. The Eisentein series contribution
  13. The cusp form contribution
  14. An application to $L$-functions
  15. Moments of $L$-functions
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $K\geq12$ be sufficiently large and $\ell\geq12$ an even integer. Assume $\ell \leq K^{\delta_2-\varepsilon}$, with $\delta_2=3/4$. Then for every $g\in H_\ell$,

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 12 more