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Simultaneous Nonvanishing of Twists of Automorphic L-Functions

Published online by Cambridge University Press:  04 December 2007

P. Michel ,
J. Vanderkam  and
P. Michel
P. Michel
Affiliation:
Mathématiques, Université Montpellier II, CC 051, 34095, Montpellier Cedex 05, France. e-mail: michel@darboux.math.univ-montp2.fr
J. Vanderkam
Affiliation:
Center for Communications Research, Thanet Road, Princeton, NJ 08540, U.S.A. e-mail: vanderkm@idaccr.org
P. Michel
Affiliation:
P.M. is partially supported by NSF Grant DMS-97-2992 and by the Ellentuck fund (by grants to the Institute for Advanced Study) and by the Institut Universitaire de France.
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Abstract

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Given three distinct primitive complex characters χ1,χ2,χ3 satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·χ1,1/2) L(f·χ2,1/2) L(f·χ3,1/2) does not vanish for a positive proportion of weight 2 primitive forms for Γ0(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J0(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.

Keywords

automorphic L-functionscentral valuesmollification
Type
Research Article
Information
Compositio Mathematica , Volume 134 , Issue 2 , November 2002 , pp. 135 - 191
Copyright
© 2002 Kluwer Academic Publishers