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On the \(p\)-adic deformations of Saito-Kurokawa lifts. (Sur les déformations \(p\)-adiques des formes de Saito-Kurokawa.) (French. Abridged English version) Zbl 1024.11030
Let \(f\) be a normalized cusp form of weight \(2k-2\) and level 1; let \(L(f,s)\) denote the associated \(L\)-function. Let \(p\) be an ordinary prime for \(f\). Denote by \(V_f\) the \(p\)-adic Galois representation associated to \(f\) by Deligne. Consider the Selmer groups \(H^1_f(\mathbb Q,V_f(n))\) \((n\in\mathbb Z)\) defined by Bloch and Kato. The authors prove that if \(L(f,s)\) vanishes at \(s=k-1\) to odd order, then \(H^1_f(\mathbb Q,V_f(k-1))\) is infinite. To prove the result they construct an extension of \(V_f\) using Galois representations associated to Siegel modular forms that are congruent modulo large powers of \(p\) to a suitable Saito-Kurokawa lift of \(f\).

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
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[1] Coleman, R.; Gouvea, F.; Jochnowitz, N., E2, θ, and overconvergence, Internat. math. res. notices, 1, 23-24, (1995)
[2] K. Kato, p-adic Hodge theory and values of zeta function of modular forms, Preprint · Zbl 1142.11336
[3] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Preprint · Zbl 1045.11029
[4] G. Laumon, Fonctions Zéta des variétés de Siegel de dimension 3, Preprint
[5] Piatetski-Shapiro, I., On the saito – kurokawa lifting, Invent. math., 71, (1983) · Zbl 0515.10024
[6] R. Schmidt, Generalized Saito-Kurokawa liftings, Preprint
[7] Tilouine, J.; Urban, E., Several variable p-adic families of siegel – hilbert cusp eigensystems and their Galois representations, Ann. sci. école norm. sup. (4), 32, 499-574, (1999) · Zbl 0991.11016
[8] E. Urban, Sur les représentations p-adiques associées aux représentations cuspidales de \(G Sp4/Q\), Preprint
[9] R. Weissauer, Four dimensional Galois representations, Preprint · Zbl 1097.11027
[10] Waldspurger, J.-L., Correspondances de Shimura et quaternions, Forum math., 3, 219-307, (1991) · Zbl 0724.11026
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