On the error term in Duke's estimate for the average special value of L-functions

Abstract

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error term of O(N^{-1/2} log N). The bound here is used in the author's paper "Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p," (to appear, Amer. J. Math.) to show that certain spaces of cuspforms arising there contain forms whose L-functions have nonvanishing special value. Version of May 2005: Nathan Ng found an error in the earlier version which yielded a bound too strong by a factor of log N; this is the corrected version, as it will appear in Canad. Math. Bull. The change does not affect the application to the Amer. J. Math. paper.