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

Abstract

Let $f$ be a newform of prime level $p$ with any central character $\ddot{I}‡(\mathrm{mod}p)$, and let $g$ be a fixed cusp form or Eisenstein series for ${SL}_2(\mathbb{Z})$. We prove the subconvexity bound: for any $\epsilon >0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\epsilon$, 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.

27 pages. Comments welcome!