On the \(p\)-adic deformations of certain automorphic representations. (Sur les déformations \(p\)-adiques de certaines représentations automorphes.) (French) Zbl 1169.11314

Skinner, Christopher; Urban, Eric

J. Inst. Math. Jussieu 5, No. 4, 629-698 (2006).

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

Keywords:

\(p\)-adic modular forms; Galois representations; Selmer groups; \(L\)-functions

Citations:

Zbl 1024.11030