1. The Burgess bound via a trivial delta method
- Author
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Qingfeng Sun, Keshav Aggarwal, Roman Holowinsky, and Yongxiao Lin
- Subjects
Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,11F66 ,0102 computer and information sciences ,Mathematical proof ,01 natural sciences ,Cusp form ,Dirichlet character ,Convolution ,Combinatorics ,Delta method ,symbols.namesake ,Number theory ,010201 computation theory & mathematics ,Fourier analysis ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - 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., Comment: 17 pages; referee comments incorporated; to appear in the Ramanujan Journal
- Published
- 2020