Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $t$-aspect

TL;DR

This paper establishes a subconvexity bound for the Rankin-Selberg $L$-function associated with $GL(3)$ and $GL(2)$ forms in the $t$-aspect, improving previous bounds and advancing understanding of automorphic $L$-functions.

Contribution

The paper proves a new subconvex bound for $L(1/2+it, \, \pi \times f)$, providing a significant improvement in the $t$-aspect for $GL(3) \times GL(2)$ $L$-functions.

Findings

Established a subconvexity bound of $(1+|t|)^{3/2 - 1/42 + \varepsilon}$.

Improved understanding of the growth of $L$-functions in the $t$-aspect.

Contributed to the analytic theory of automorphic forms and $L$-functions.

Abstract

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb Z)$ and $f$ be a holomorphic (or Maass) Hecke form for $SL(2,\mathbb{Z})$. In this paper we prove the following subconvex bound $$ L\left(\tfrac{1}{2}+it,\pi\times f\right)\ll_{\pi,f,\varepsilon} (1+|t|)^{\frac{3}{2}-\frac{1}{42}+\varepsilon}. $$