A Bessel $\delta$-method and hybrid bounds for $\mathrm{GL}_2$

TL;DR

This paper introduces a Bessel δ-method and derives hybrid bounds for L-functions associated with automorphic forms, improving previous subconvexity bounds using novel integral identities and asymptotic analysis.

Contribution

The paper develops a new Bessel δ-identity for automorphic forms and applies it to establish improved hybrid subconvexity bounds for L-functions.

Findings

Proves an asymptotic Bessel δ-identity for automorphic forms.

Establishes a hybrid subconvexity bound for L(1/2+it, g⊗χ).

Improves previous bounds on L-functions for primitive forms and Dirichlet characters.

Abstract

Let $g$ be a primitive holomorphic or Maass newform for $\Gamma_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $\delta$-identity associated to $g$. Among other applications, we prove the following hybrid subconvexity bound $$ L\left(1/2+it,g\otimes \chi\right)\ll_{g,\varepsilon} (q(1+|t|))^{\varepsilon}q^{3/8}(1+|t|)^{1/3} $$ for any $\varepsilon>0$, where $\chi \bmod q$ is a primitive Dirichlet character with $(q, D)=1$. This improves the previous known result.