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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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