Special values of anticyclotomic L-functions

The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the L-functions associated to modular forms in the anticyclotomic tower of conductor p^(infinity) over an imaginary quadratic field. While the authorʹs previous work proved that such L-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is +1, the present work includes the case where the sign is -1. In that case, it is shown that the derivatives of the Lfunctions are generically nonzero at the center. It is also shown that when the sign is +1, the algebraic part of the central critical value is nonzero modulo l for certain l Applications are given to the mu-invariant of the p-adic L-functions of M. Bertolini and H. Darmon. The main ingredients in the proof are a theorem of M. Ratner, as in the authorʹs previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.