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\(p\)-adic periods and modular symbols of elliptic curves of prime conductor. (English) Zbl 1044.11576
Summary: Under certain assumptions, the author proves a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction at \(p\), and its \(p\)-period. He generalizes this relation to modular forms of weight 2 with coefficients not necessarily in \(\mathbb Q\).

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F33 Congruences for modular and \(p\)-adic modular forms
11F55 Other groups and their modular and automorphic forms (several variables)
11G05 Elliptic curves over global fields
11G20 Curves over finite and local fields
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
Full Text: DOI EuDML
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