# Subconvexity for $GL(3)\times GL(2)$ twists (with an appendix by Will Sawin)

## TL;DR

This paper establishes new subconvex bounds for certain $L$-functions involving $GL(3)$ and $GL(2)$ automorphic forms twisted by characters, advancing understanding of their growth rates at the critical line.

## Contribution

It provides the first subconvexity bounds for $L$-functions of $GL(3) imes GL(2)$ twists, improving previous convexity bounds and employing novel analytic techniques.

## Key findings

- Proves $L(1/2, \pi imes f imes \chi) ext{ bound } p^{3/2 - 1/16 + ext{small }\epsilon}$.
- Establishes $L(1/2, \pi imes \chi) ext{ bound } p^{3/4 - 1/32 + ext{small }\epsilon}$.
- Advances the analytic theory of automorphic $L$-functions and their subconvexity bounds.

## Abstract

Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass cusp form and $\chi$ be any non-trivial character $\bmod \, p$, where $p$ is prime. We show that the $L$-function associated with this triplet satisfy \begin{equation*} L\left(\frac{1}{2},\pi\times f\times\chi\right)\ll_{\pi,f,\epsilon} p^{\frac{3}{2}-\frac{1}{16}+\epsilon}. \end{equation*} The method also yields the subconvex bound \begin{equation*} L\left(\frac{1}{2},\pi\otimes \chi\right)\ll_{\pi,\epsilon }p^{\frac{3}{4}-\frac{1}{32}+\epsilon }. \end{equation*}