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On the $$p$$-adic deformations of certain automorphic representations. (Sur les déformations $$p$$-adiques de certaines représentations automorphes.) (French) Zbl 1169.11314
Summary: By an entirely new method that makes use of $$p$$-adic deformations of automorphic representations of $$\mathrm{GSp}_{4}/\mathbb{Q}$$, we prove that the $$p$$-adic Selmer group $$H^1_f(\mathbb{Q},V_f(k))$$ associated to a modular form $$f$$ of weight $$2k$$ that is ordinary at $$p$$ is infinite if the order of vanishing at $$k$$ of the $$L$$-function of $$f$$ is odd.
See also the authors’ announcement in C. R., Math., Acad. Sci. Paris 335, No. 7, 581–586 (2002; Zbl 1024.11030).

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 11F85 $$p$$-adic theory, local fields 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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