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Special values of anticyclotomic $$L$$-functions. (English) Zbl 1065.11048
The author extends his earlier results and methods [V. Vatsal, Invent. Math. 148, No. 1, 1–46 (2002; Zbl 1119.11035)] on nonvanishing of $$L$$-functions associated to modular forms in the anticyclotomic tower of conductor $$p^\infty$$ over an imaginary quadratic field to the case where the sign in the functional equation is $$-1$$ (Theorem 1.4). In that case he shows, in particular, that derivatives of the $$L$$-functions are generically nonzero at the center. The main ingredient in the proof is a new “Johnowitz congruence” that relates the nontriviality of a special value modular curve.
Applications are given to the Iwasawa $$\mu$$-invariant of the $$p$$-adic $$L$$-functions of Bartolini and Darmon (Theorem 1.1).

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F33 Congruences for modular and $$p$$-adic modular forms 11G18 Arithmetic aspects of modular and Shimura varieties 11R23 Iwasawa theory
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