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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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##### References:
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