On p-adic families of Hilbert cusp forms of finite slope

Abstract

Let p be a prime number and F a totally real field. In this article, we obtain a p-adic interpolation of spaces of totally definite quaternionic automorphic forms over F of finite slope, and construct p-adic families of automorphic forms parametrized by affinoid Hecke varieties. Further, as an application to the case where [F : Q] is even, we obtain p-adic analytic families of Hilbert eigenforms having fixed finite slope parametrized by weights. This is an analogue of Coleman's analytic families in [R.F. Coleman, p-Adic Banach spaces and families of modular forms, Invent. Math. 127 (1997) 417-479].

Overconvergent algebraic automorphic forms

“…As a final example, we consider the case of quaternionic Hilbert modular forms studied in [30] and [8,part III]. Let F be a totally real field and B a totally definite quaternion algebra over F (i.e.…”

“…Relation to other constructions. As mentioned above, the methods of this paper are a generalisation of those used in several earlier works dealing with specific examples of compact-modulo-centre reductive groups; the main examples are [7], [8], [10] and [30]. This is considered in detail in section 4, where we show how to recover some of the results of these papers from our approach.…”

Families of Galois representations and Selmer groups

“…The choice of a quadratic imaginary field in this chapter rather than a general CM field (as well as the choice of a split p) is made mainly to simplify the exposition and also because this the only case that we shall use in the applications to Selmer groups (see §9.5.1). All the constructions actually extend to this more general setting by combining the arguments here (or of [36]) and those of [32] (see also Yamagami's work [124]). Alternatively, the general definite case is now covered by Emerton's paper [53].…”

Classical forms of weight one in ordinary families

“…Partially ordinary families are also discussed briefly in [28], again working cohomologically with quaternionic automorphic forms. We should mention that similar families have been constructed independently by Yamagami in [31] and by Johansson-Newton in [19]. Both Yamagami and Johansson-Newton work with more general v-finite-slope rather than v-ordinary families, though one can think of the ord p (U v ) = 0 locus in the v-finite-slope eigenvariety as being the v-ordinary families that we study.…”