The Burgess bound via a trivial delta method
- PDF / 448,233 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 3 Downloads / 139 Views
The Burgess bound via a trivial delta method Keshav Aggarwal1 · Roman Holowinsky1
· Yongxiao Lin1 · Qingfeng Sun2
Received: 22 June 2018 / Accepted: 6 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let g be a fixed Hecke cusp form for SL(2, Z) and χ be a primitive Dirichlet character of conductor M. The best known subconvex bound for L(1/2, g ⊗ χ ) 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 GL(2). In this paper, we give a new proof of the Burgess-type bounds L(1/2, g ⊗ χ ) g,ε M 1/2−1/8+ε and L(1/2, χ ) ε M 1/4−1/16+ε that does not require the basic tools of the previous proofs and instead uses a trivial delta method. Keywords Subconvexity · Dirichlet characters · Hecke cusp forms · L-functions Mathematics Subject Classification 11F66
1 Introduction and statement of results Let g be a fixed Hecke cusp form on GL(2) and let χ be a primitive Dirichlet character modulo M. Subconvex bounds for the twisted L-functions
Q. S. was partially supported by NSFC (Grant No. 11871306) and CSC.
B
Roman Holowinsky [email protected] Keshav Aggarwal [email protected] Yongxiao Lin [email protected] Qingfeng Sun [email protected]
1
Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210-1174, USA
2
School of Mathematics and Statistics, Shandong University, Weihai 264209, Shandong, China
123
K. Aggarwal et al.
L(s, g ⊗ χ ) =
∞ λg (n)χ (n) n=1
ns
have been of interest for a while, with lots of applications. The strongest known bound is of Burgess quality (in special cases better results than Burgess-type exponent are known). The first subconvex bound in the conductor aspect was established by Burgess [4] for Dirichlet L-functions L(s, χ ). For a primitive Dirichlet character χ modulo M, Burgess proved 1 , χ ε M 1/4−1/16+ε . (1) L 2 A subconvex bound of such strength is called a Burgess bound. In the GL(2) setting, the Burgess bound for an L-function of a Hecke cusp form g twisted by a primitive Dirichlet character χ of large conductor M is L
1 ,g ⊗χ 2
g,ε M 1/2−1/8+ε .
(2)
The first subconvex bound for L (1/2, g ⊗ χ ) was obtained by Duke et al. [7], who studied an amplified second moment and reduced the problem to treating shifted convolution sums of the form
λg (m) λg (n),
1 m−2 n=h
for which they invented the δ-symbol method in their name to deal with. Their approach gives 1 , g ⊗ χ g,ε M 1/2−1/22+ε L 2 where g is a holomorphic cusp form on the full modular group. Bykovski˘ı [5], who embedded the L-function L (1/2, g ⊗ χ ) in question into a larger family 2 g ∈S(M) L(1/2, g ⊗ χ ) (and with amplification) and treated the later by applying the Petersson formula, where S(M) is an orthogonal basis of the space of holomorphic cusp forms of level M, was able to sharpen the bound of Duke et al. to
Data Loading...
