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Hybrid subconvexity bounds for twists of $\rm GL_2\times\rm GL_2$ $L$-functions

Chenchen Shao, Huimin Zhang

TL;DR

This work proves a hybrid subconvexity bound for twists of GL$_2\times$GL$_2$ Rankin–Selberg L-functions by a primitive Dirichlet character modulo a prime power, in both the $t$- and depth-aspects. The authors combine the approximate functional equation with a conductor-lowering delta method, GL$_2$ Voronoi summation, and a Cauchy–Schwarz–Poisson cascade to extract savings, optimizing parameters to achieve $L\left(\tfrac{1}{2}+it,f\times g\times \chi\right) \ll_{\varepsilon} p^{\frac{3}{4}}(p^{\kappa})^{1-\frac{1}{16}+\varepsilon}(|t|+1)^{1-\frac{1}{10}+\varepsilon}$. Key innovations include dual-length analysis via stationary phase, a refined delta method in the depth aspect, and careful handling of zero and non-zero frequencies across large and small modulo regimes. The result advances hybrid subconvexity in the GL$_2\times$GL$_2$ setting and broadens the applicability of delta-method techniques to simultaneous depth and $t$-aspects with prime-power conductors.

Abstract

In this paper, we prove hybrid subconvexity bounds for $\rm GL_2\times \rm GL_2$ Rankin--Selberg $L$-functions twisted by a primitive Dirichlet character $χ$ modulo a prime power, in the $t$ and depth aspects.

Hybrid subconvexity bounds for twists of $\rm GL_2\times\rm GL_2$ $L$-functions

TL;DR

This work proves a hybrid subconvexity bound for twists of GLGL Rankin–Selberg L-functions by a primitive Dirichlet character modulo a prime power, in both the - and depth-aspects. The authors combine the approximate functional equation with a conductor-lowering delta method, GL Voronoi summation, and a Cauchy–Schwarz–Poisson cascade to extract savings, optimizing parameters to achieve . Key innovations include dual-length analysis via stationary phase, a refined delta method in the depth aspect, and careful handling of zero and non-zero frequencies across large and small modulo regimes. The result advances hybrid subconvexity in the GLGL setting and broadens the applicability of delta-method techniques to simultaneous depth and -aspects with prime-power conductors.

Abstract

In this paper, we prove hybrid subconvexity bounds for Rankin--Selberg -functions twisted by a primitive Dirichlet character modulo a prime power, in the and depth aspects.

Paper Structure

This paper contains 28 sections, 16 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Preliminaries
  4. The delta method
  5. Holomorphic cusp forms for $\mathrm{GL}_2$
  6. Maass cusp forms for $\mathrm{GL}_2$
  7. Estimates for exponential integrals
  8. Setting
  9. Approximate functional equation
  10. Applying DFI's delta method
  11. Voronoi summation formulas
  12. Applying Cauchy--Schwarz inequality and Poisson summation formula
  13. The large modulo case
  14. The small modulo case
  15. The zero frequency
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $p^{\kappa}$ with p an odd prime and $\kappa >12$. The following hybrid subconvexity bound holds:

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Remark 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 14 more