A Bessel $\hat{I}´$-method and hybrid bounds for ${\mathrm{GL}}_2$

Abstract

Let $g$ be a primitive holomorphic or Maass newform for $\hat{I}{“}_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $\hat{I}´$-identity associated to $g$. Among other applications, we prove the following hybrid subconvexity bound $$L\left(1/2+it,g\otimes \ddot{I}‡\right){\ll }_{g,\epsilon }(q(1+|t|){)}^{\epsilon }{q}^{3/8}(1+|t|{)}^{1/3}$$ for any $\epsilon >0$, where $\ddot{I}‡\mathrm{mod}q$ is a primitive Dirichlet character with $(q,D)=1$. This improves the previous known result.

30 pages