On the growth of Rankin-Selberg L-functions for $SL(2)$

TL;DR

This paper derives new bounds for Rankin-Selberg L-functions for SL(2) by leveraging Eisenstein series supnorms and representation theory, achieving a subconvexity bound for Maass cusp forms.

Contribution

It introduces a novel approach combining Eisenstein series supnorm estimates and representation theory to bound L-functions, leading to a subconvexity result.

Findings

Established subconvexity bound for L(1/2 + it, f1 x f2)

Connected Eisenstein series supnorms with L-function bounds

Applied representation theory to analytic number theory problems

Abstract

In this paper, we establish bounds of the Rankin-Selberg $L$-function for $SL(2)$ using the supnorm of the Eisenstein series and a purely representation theoretic index over the real group. Consequently, we obtain a subconvexity bound $L(\frac{1}{2}+ it, f_1 \times f_2) \leq C (1+ |t|)^{\frac{5}{6}+\epsilon}$ for two Maass cusp forms of $SL(2, \mathbb Z)$.