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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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