Special values of anticyclotomic $L$-functions

Abstract

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 $p\sp \infty$ over an imaginary quadratic field. While the author's 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 $L$-functions 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 $\ell$ for certain $\ell$. 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 author's previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.