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Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields

Asbjorn Christian Nordentoft

Abstract

We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(π\otimes Ω, 1/2)$ where $π$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $Ω$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain "weak simultaneous" non-vanishing results, which are non-vanishing results for different Rankin--Selberg $L$-functions where the product of the twists is trivial. The proof relies on relating the wide moments to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger's formula. To achieve this, a classical version of Waldspurger's formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error-terms) together with non-vanishing results for certain period integrals. In particular, we develop a soft technique for obtaining non-vanishing of triple convolution $L$-functions.

Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields

Abstract

We calculate certain "wide moments" of central values of Rankin--Selberg -functions where is a cuspidal automorphic representation of over and is a Hecke character (of conductor ) of an imaginary quadratic field. This moment calculation is applied to obtain "weak simultaneous" non-vanishing results, which are non-vanishing results for different Rankin--Selberg -functions where the product of the twists is trivial. The proof relies on relating the wide moments to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger's formula. To achieve this, a classical version of Waldspurger's formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error-terms) together with non-vanishing results for certain period integrals. In particular, we develop a soft technique for obtaining non-vanishing of triple convolution -functions.

Paper Structure

This paper contains 29 sections, 20 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Wide moments of $L$-functions
  3. Main idea
  4. Non-vanishing results
  5. Main moment calcualtion
  6. Weak simultaneous non-vanishing
  7. Background
  8. Different incarnations of the class group
  9. Heegner forms
  10. Oriented embeddings
  11. Hecke characters of imaginary quadratic fields
  12. Automorphic forms
  13. Adélization of Maaß forms
  14. Automorphic $L$-functions
  15. A classical version of Waldspurger's formula
  16. ...and 14 more sections

Key Result

Corollary 1.1

Let $f$ be either a Hecke--Maaß cusp form of spectral parameter $t_f$ and level $1$ or a cuspidal holomorphic Hecke eigenform of weight $k_f$ and level $1$. Let $k$ be a positive even integer such that $k\geq k_f$ when $f$ is holomorphic. Put $T=|t_f|+k+1$ in the Maaß case and $T=k+1$ in the holomor where $\Omega_K$ is a Hecke character of $K$ of conductor $1$ and $\infty$-type $\alpha\mapsto (\al

Theorems & Definitions (39)

  • Corollary 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • proof
  • Lemma 4.1
  • ...and 29 more