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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Editorial Board Member: Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Strambach, Karl


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A Burgess-like subconvex bound for twisted L-functions

Citation Information: Forum Mathematicum. Volume 19, Issue 1, Pages 61–105, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/FORUM.2007.003, February 2007

Publication History

Received:
2004-11-23
Published Online:
2007-02-21

Abstract

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = ½. It is proved that

, where ε > 0 is arbitrary and θ = is the current known approximation towards the Ramanujan–Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L-functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.

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