Hilbert modular forms: mod 𝑝 and 𝑝-adic aspects

Abstract

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.

A theta operator on Picard modular forms modulo an inert prime

“…j(γ ; z, u)) is trivial on Γ cusp , the stabilizer of c ∞ in Γ , so dζ 1 ∧ dζ 2 and dζ 3 define sections of det P and L on S C , the formal completion ofS C along the cuspidal divisor E c = p −1 (c ∞ ) ⊂S C . The same also holds for dζ 1 and dζ 2 mod dζ 1 individually (Lemma 1.8). Along E c , P has a canonical filtration…”

“…This allowed us to define sections dζ 1 , dζ 2 and dζ 3 of ω A/X . A simple matrix computation gives the following.…”

“…1.2.4. Here dζ 1 and dζ 2 form a basis for P and dζ 3 for L. The same coordinates served to define also the semi-abelian variety G u (denoted also A u ) over the cuspidal component E at c ∞ , cf Sect. 1.6.…”

“…To explain how we overcome the need to consider the unit-root splitting of the cohomology of the universal abelian variety, let us re-examine the case of the modular curve X of full level N ≥ 3 over Z p ((p, N ) = 1). We follow an approach of Gross [13], see also [1], who extended it to Hilbert modular varieties. Let κ be a fixed algebraic closure of F p , and consider the geometric characteristic p fiber X κ .…”

“…Let A be the universal semi-abelian variety overS. It is relatively 3-dimensional, has complex multiplication by O K , and the cotangent bundle at the origin, ω A/S , is of type (2,1). This means that if Σ : O K → R 0 is the canonical embedding andΣ its complex conjugate, then…”