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arXiv · 1803.00542

The Burgess bound via a trivial delta method

Keshav Aggarwal·Roman Holowinsky·Yongxiao Lin·Qingfeng Sun
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Abstract

Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of conductor $M$. The best known subconvex bound for $L(1/2,g\otimes \chi)$ 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 $\rm GL(2)$. In this paper, we give a new proof of the Burgess-type bounds ${L(1/2,g\otimes \chi)\ll_{g,\varepsilon} M^{1/2-1/8+\varepsilon}}$ and $L(1/2,\chi)\ll_{\varepsilon} M^{1/4-1/16+\varepsilon}$ that does not require the basic tools of the previous proofs and instead uses a trivial delta method.

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Keshav Aggarwal, Roman Holowinsky, Yongxiao Lin, Qingfeng Sun. 2018-03-01. The Burgess bound via a trivial delta method. https://arxiv.org/abs/1803.00542

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