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The cubic moment of $L$-functions for specified local component families

Yueke Hu, Ian Petrow, Matthew P. Young

Abstract

We prove Lindelöf-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $π$ given by specifying the local representation $π_p$ of $π$ at finitely many primes. Such bounds were previously known in the case that $π_p$ belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of $\operatorname{PGL}_2$ $L$-functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such a bound was only known for forms that are twists from level $p$, which cover roughly half of the level $p^2$ forms.

The cubic moment of $L$-functions for specified local component families

Abstract

We prove Lindelöf-on-average upper bounds on the cubic moment of central values of -functions over certain families of automorphic representations given by specifying the local representation of at finitely many primes. Such bounds were previously known in the case that belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of -functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level . Previously, such a bound was only known for forms that are twists from level , which cover roughly half of the level forms.

Paper Structure

This paper contains 34 sections, 54 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Related works
  4. Overview of the proof
  5. Acknowledgements
  6. Background tools
  7. The PBK formula
  8. Local Fields
  9. Neighborhood of a local representation
  10. Specified local component family
  11. The PBK formula for specified local components
  12. Kloosterman sum properties
  13. Postnikov formula
  14. The multivariable Postnikov formula over $\mathbb Q_p$
  15. Coset fourth moment
  16. ...and 19 more sections

Key Result

Theorem 1.1

There exists $B>0$ such that for any local component family $\mathcal{F}$ as in Section sec:familydef, we have $\mathcal{M}(\mathcal{F}) \ll_\varepsilon |\Pi_\infty|^{B} \prod_p p^{\lceil \frac{\alpha_p}{2}\rceil(1+\varepsilon)}$. If there exists $\delta>0$ such that $|s| \geq Q^\delta$ for all $s\i

Theorems & Definitions (102)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2: PBK formula for specified local components
  • proof
  • Lemma 2.3
  • Lemma 2.4: Postnikov
  • Lemma 2.5
  • proof
  • ...and 92 more