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Toroidal families and averages of $L$-functions, II: cubic moments

Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, Will Sawin

Abstract

Generalizing our previous work on ``toroidal averages'', we study the average of special values of $L$-functions of the form $L(1/2,χ^a)L(1/2,χ^b)L(1/2,χ^c)$ for integers $a$, $b$ and $c$, where $χ$ varies over Dirichlet characters of a given prime modulus. We highlight connections with estimates for bilinear forms of trace functions and with bounds for the number of solutions of monoidal equations in three variables in small boxes over finite fields.

Toroidal families and averages of $L$-functions, II: cubic moments

Abstract

Generalizing our previous work on ``toroidal averages'', we study the average of special values of -functions of the form for integers , and , where varies over Dirichlet characters of a given prime modulus. We highlight connections with estimates for bilinear forms of trace functions and with bounds for the number of solutions of monoidal equations in three variables in small boxes over finite fields.
Paper Structure (48 sections, 33 theorems, 373 equations)

This paper contains 48 sections, 33 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Overview of the proof of Theorem
  4. Structure of the paper
  5. Bounds for multilinear sums of trace functions
  6. Bounds for bilinear sums of trace functions
  7. The hypergeometric sheaves for the cubic moment
  8. Statement of the result
  9. Construction of a complex
  10. Hypergeometric representation
  11. The positive case
  12. The negative case
  13. Analysis of the cubic mixed moment
  14. Reduction to the coprime case
  15. Approximate functional equations for triples of Dirichlet $L$-functions
  16. ...and 33 more sections

Key Result

Theorem 1.2

Let $a,b,c\geqslant 1$ be setwise coprime integers, $d\geqslant 1$ another integer and $q\geqslant 3$ be a prime. Assume that $(a,b,\pm c)$ is galant or oxozonic. Furthermore we have $D_{a,b,c}=1$ and $D_{a,b,-c}=D_{a,b,-c}(1/2)$ is the value ($>1$) at $s=1/2$ of the converging Dirichlet series $D_{a,b,-c}(s)$ defined in defDabc.

Theorems & Definitions (66)

  • Definition 1.1: Galant triples
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.2
  • Theorem 1.5
  • Remark 1.3
  • Remark 1.4
  • Theorem
  • ...and 56 more