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Exact double averages of twisted L-values

Kimball Martin

TL;DR

The paper advances the study of central L-values twisted by CM characters by proving exact double-average formulas over weight-2 forms and ideal-class characters, using a quaternionic framework with Brandt matrices and the Jacquet--Langlands correspondence. It generalizes prior stable single-average results (MR, FW) to broader level structures, including non-squarefree and even-prime levels, and accommodates various ramification patterns of $K$. The approach yields explicit stability criteria, effective nonvanishing bounds, and consequences over $\mathbb{Q}$, along with a robust apparatus for varying both level and CM field. This contributes a versatile toolset for understanding nonvanishing and distribution of twisted L-values in families of cusp forms.

Abstract

Consider central $L$-values of even weight elliptic or Hilbert modular forms $f$ twisted by ideal class characters $χ$ of an imaginary quadratic extension $K$. Fixing $χ$, and assuming $K$ is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms $f$ of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to $K$ (the "stable range"). In weight 2, we obtain exact formulas for a simultaneous average over both $f$ and $χ$. We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of $K$ above the level. Under elementary conditions on the level, these double averages are "stable" in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central $L$-values.

Exact double averages of twisted L-values

TL;DR

The paper advances the study of central L-values twisted by CM characters by proving exact double-average formulas over weight-2 forms and ideal-class characters, using a quaternionic framework with Brandt matrices and the Jacquet--Langlands correspondence. It generalizes prior stable single-average results (MR, FW) to broader level structures, including non-squarefree and even-prime levels, and accommodates various ramification patterns of . The approach yields explicit stability criteria, effective nonvanishing bounds, and consequences over , along with a robust apparatus for varying both level and CM field. This contributes a versatile toolset for understanding nonvanishing and distribution of twisted L-values in families of cusp forms.

Abstract

Consider central -values of even weight elliptic or Hilbert modular forms twisted by ideal class characters of an imaginary quadratic extension . Fixing , and assuming is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to (the "stable range"). In weight 2, we obtain exact formulas for a simultaneous average over both and . We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of above the level. Under elementary conditions on the level, these double averages are "stable" in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central -values.

Paper Structure

This paper contains 27 sections, 29 theorems, 92 equations, 1 table.

Table of Contents

  1. Introduction
  2. Prime power level
  3. A more general class of levels
  4. Methods
  5. Related work
  6. Notation
  7. Quaternionic modular forms
  8. Special orders
  9. Automorphic forms
  10. Brandt matrices and representation numbers
  11. Jacquet--Langlands correspondence
  12. Average values of quaternionic modular forms
  13. Periods and embeddings
  14. Periods and averages
  15. Exact averages and stability
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose $N = p^{2r+1} \ge 11$ for a prime $p$ with $N \ne 27$, and $K/\mathbb Q$ is an imaginary quadratic field which is not split at $p$. If $r > 0$, assume $K/\mathbb Q$ is inert at $p$. Put where $e_p(K/\mathbb Q) = 1$ if $K/\mathbb Q$ is inert at $p$ and $e_p(K/\mathbb Q)=2$ if $K/\mathbb Q$ is ramified at $p$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 3.1: HPS
  • Lemma 3.2
  • proof
  • ...and 53 more