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Forum Mathematicum

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Volume 19, Issue 1 (Jan 2007)


A Burgess-like subconvex bound for twisted L-functions

V Blomer
  • Mathematisches Institut, Bunsenstrasse 3-5, D-37073 Göttingen, Germany.
  • Email:
/ G Harcos
  • The University of Texas at Austin, Mathematics Department, 1 University Station C1200, Austin, TX 78712-0257, USA.
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/ P Michel
  • I3M, UMR CNRS 5149, Université Montpellier II CC 051, 34095 Montpellier Cedex 05, France.
  • Email:
/ Z Mao
  • Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102-1811, USA.
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Published Online: 2007-02-21 | DOI: https://doi.org/10.1515/FORUM.2007.003


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.

About the article

Received: 2004-11-23

Published Online: 2007-02-21

Published in Print: 2007-01-29

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/FORUM.2007.003.

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