The Burgess bound via a trivial delta method

Abstract

Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $\ddot{I}‡$ be a primitive Dirichlet character of conductor $M$. The best known subconvex bound for $L(1/2,g\otimes \ddot{I}‡)$ is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type bound on $\mathrm{GL}(2)$. In this paper, we give a new proof of the Burgess-type bounds $L(1/2,g\otimes \ddot{I}‡){\ll }_{g,\epsilon }{M}^{1/2-1/8+\epsilon }$ and $L(1/2,\ddot{I}‡){\ll }_{\epsilon }{M}^{1/4-1/16+\epsilon }$ that does not require the basic tools of the previous proofs and instead uses a trivial delta method.

17 pages; referee comments incorporated; to appear in the Ramanujan Journal