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Subconvexity Problem on $\operatorname{GL}_3$ over number fields: the twist aspect

Filippo Berta

Abstract

Let $F$ denote a number field and let $\mathfrak{q}\subset O_F$ traverse a sequence of prime ideals with norm $N(\mathfrak{q}) \to \infty$ and for each $\mathfrak{q}$, let $χ\in \widehat{F^{\times}\setminus \mathbb{A}^\times}$ be a finite order character of conductor $\mathfrak{q}$. For a fixed unitary cuspidal automorphic representation $π$ of $\operatorname{GL}_3/F$ we show that \begin{equation*} L(π\otimes χ,\tfrac{1}{2})\ll \ N(\mathfrak{q})^{3/4-κ}.\end{equation*} holds for all $κ< \frac{1}{36}$.

Subconvexity Problem on $\operatorname{GL}_3$ over number fields: the twist aspect

Abstract

Let denote a number field and let traverse a sequence of prime ideals with norm and for each , let be a finite order character of conductor . For a fixed unitary cuspidal automorphic representation of we show that \begin{equation*} L(π\otimes χ,\tfrac{1}{2})\ll \ N(\mathfrak{q})^{3/4-κ}.\end{equation*} holds for all .
Paper Structure (30 sections, 27 theorems, 262 equations)

This paper contains 30 sections, 27 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Strategy for the proof and structure of the paper
  3. Notation and basic set-up
  4. Asymptotic notation
  5. Choice of fundamental domain for $F^{\times}\setminus\mathbb{A}^{(1)}$
  6. Special elements
  7. Additive character
  8. Measures
  9. Newforms
  10. Counting units
  11. Gauss sums and Kloosterman sums
  12. Gauss sums
  13. Kloosterman sums
  14. Correlation of Kloosterman sums in smooth boxes
  15. Automorphic representations and integral representation of $L$-functions
  16. ...and 15 more sections

Key Result

Theorem 1.1

The subconvex bound eq:Theorem holds for any $\kappa_0 < \frac{1}{36},$ provided that the ideal conductor $\mathfrak{C}(\pi)$ of $\pi$ is coprime to the discriminant ideal of $F$. Moreover, the implicit constant depends at most polynomially on $\mathfrak{C}(\pi)$ and on the invariants of $F$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1: Dirichlet's unit theorem
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 42 more