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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].

##### MSC:
 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