p-adic families of Siegel modular cuspforms

Abstract

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.

On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image

“…The aim of this paper is to study the same question for Hilbert modular forms i.e. to study the geometry of the eigenvariety for Hilbert modular forms constructed by Andreatta-Iovita-Pilloni in [1] at classical, regular points of parallel weight 1.…”

“…In [1], Andreatta, Iovita and Pilloni constructed the eigenvariety E for cuspidal Hilbert modular eigenforms of tame level N defined over F with the Hecke operators U (p i ) for i = 1, · · · , r and T (q) for primes q N p. The normalization of the operators U (p i ) used in [1] to construct E is different from the classical normalization. The normalization that they use is the same as the one defined by Hida in [13].…”

“…Note that these operators coincide with the classical U (p i )'s on parallel weight Hilbert modular forms. See remark 4.7 of [1] and [2,Section 3] for more details. Let T be Res O F /Z G m and M be a finite extension of Q p which splits F .…”

“…Let T be Res O F /Z G m and M be a finite extension of Q p which splits F . There exists a locally finite map κ : E → W, where the weight space W is the rigid analytic space over M associated to the completed group algebra O M [[T(Z p ) × Z × p ]] (see [1,Section 2] for more details). In the ordinary case, their construction is same as the construction of Hida families using Katz's p-adic modular forms.…”

“…The construction of Andreatta, Iovita and Pilloni in the parallel weight case is equivalent to Kisin-Lai's construction (see [1,Section 1]). So, if we denote the weights of E by (ν, w) following [1], then we can identify C cusp , the cuspidal part of the Kisin-Lai eigencurve C, with the closed subspace of E obtained after making ν = 0.…”