Simultaneous Nonvanishing of Twists of Automorphic L -Functions

Given three distinct primitive complex characters χ 1 ,χ 2 ,χ 3 satisfying some technical conditions, we prove that the triple product of twisted L -functions L ( f ·χ 1 ,1/2) L ( f ·χ 2 ,1/2) L ( f ·χ 3 ,1/2) does not vanish for a positive proportion of weight 2 primitive forms for Γ 0 ( q ), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J 0 ( q ) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.