Let p be an odd prime and g ≥ 2 an integer. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this theorem relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.

Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

In this paper we will give explicit descriptions of Hyodo and Kato's Frobenius and Monodromy operators on the first p-adic de Rham cohomology groups of curves and Abelian varieties with semi-stable reduction over local fields of mixed characteristic. This paper was motivated by the first author's paper [C-pSI], where conjectural definitions of these operators for curves with semi-stable reduction were given. Although B.LeStum wrote a paper entitled "La structure de Hyodo-Kato pour les courbes" ([LS]) he did not prove or claim to have proved that the operators he defined there were the same as Hyodo and Kato's.The paper is naturally divided into two chapters. In Chapter I, written by the

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In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and Perrin-Riou [16], we define restricted Selmer groups and λ ± , µ ± -invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic Zp-extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified Zp-extension of Qp.