Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions

TL;DR

This paper establishes a new subconvexity bound for certain $L$-functions associated with $ extrm{GL(2)} imes extrm{GL(2)}$, improving previous results by employing advanced bounds on bilinear forms in Kloosterman fractions.

Contribution

The paper introduces an improved subconvexity bound for $L(1/2, f imes g)$ with prime level $p$, utilizing novel bounds on bilinear forms in Kloosterman fractions.

Findings

Proves $L(1/2, f imes g) \, \ll p^{1/2 - 1/524 + \varepsilon}$ for prime level $p$.

Improves upon the previous best-known subconvexity results.

Employs advanced techniques in bounding bilinear forms in Kloosterman fractions.

Abstract

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee.