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.
We study Hilbert modular forms in characteristic p and over padic rings. In the characteristic p-theory we describe the kernel and image of the q-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators U , V and Θχ and study the variation of the filtration under these operators. In particular, we prove that every ordinary eigenform has filtration in a prescribed box of weights. Our methods are geometriccomparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-p structure, whose poles are supported on the nonordinary locus.In the p-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define p-adic Hilbert modular forms "à la Serre" as p-adic uniform limit of classical modular forms, and compare them with the p-adic modular forms "à la Katz" that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators V and Θχ to the p-adic setting.
Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n, 2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions.As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
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 $\leq h$.
Contents 1.2 NotationsLet p > 0 be a prime integer, O K a complete discrete valuation ring with fraction field K and perfect residue field k. Fix an algebraic closure K ⊂ K, let k denote its residue field and O K the normalization of O K in K. Write G K for the Galois group of K over K. Fix a field extension K ⊂ M ⊂ K. We write M 0 ⊆ M for the maximal absolutely unramified subfield of M and O M 0 for its ring of integers.The following notations will be used throughout the paper (some of the objects denoted here will be defined in this very section and the rest in the next sections):
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