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\(p\)-adic Hodge theory and values of zeta functions of modular forms. (English) Zbl 1142.11336
Berthelot, Pierre (ed.) et al., Cohomologies \(p\)-adiques et applications arithmétiques (III). Paris: Société Mathématique de France (ISBN 2-85629-118-9/pbk). Astérisque 295, 117-290 (2004).
Summary: If \(f\) is a modular form, we construct an Euler system attached to \(f\) from which we deduce bounds for the Selmer groups of \(f\). An explicit reciprocity law links this Euler system to the \(p\)-adic zeta function of \(f\) which allows us to prove a divisibility statement towards Iwasawa’s main conjecture for \(f\) and to obtain lower bounds for the order of vanishing of this \(p\)-adic zeta function. In particular, if \(f\) is associated to an elliptic curve \(E\) defined over \(\mathbb Q \), we prove that the \(p\)-adic zeta function of \(f\) has a zero at \(s=1\) of order at least the rank of the group of rational points on \(E\).
For the entire collection see [Zbl 1052.00008].

11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G35 Modular and Shimura varieties