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Licensed Unlicensed Requires Authentication Published by De Gruyter February 21, 2007

A Burgess-like subconvex bound for twisted L-functions

  • V Blomer EMAIL logo , G Harcos , P Michel and Z Mao
From the journal Forum Mathematicum

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.


(Communicated by Peter Sarnak)


Received: 2004-11-23
Published Online: 2007-02-21
Published in Print: 2007-01-29

© Walter de Gruyter

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