Slopes of Overconvergent Hilbert Modular Forms

Abstract

We give an explicit description of the matrix associated to the Up operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary of weight space for certain real quadratic fields. Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the Up.

2‐adic slopes of Hilbert modular forms over Q(5)

“…This theorem is in line with [3, Conjecture 4.7.9], appropriately generalized to this situation where the level is not sufficiently small . The proof is essentially an extended exercise on ‐adic matrix analysis which we do not know how to generalize to many other situations.…”

“…The above theorem then says that once we know a small set of classical slopes in each component of weight space, then we can obtain all slopes in any component of weight space. Remark (1)The condition that is odd is required in order for the relevant space of overconvergent Hilbert modular forms to be non‐trivial. (2)Note that this is a simple generalization of [3, Conjectures 4.7.1 and 4.7.8] to levels which are not sufficiently small (meaning we have is non‐trivial). Having a level which is not sufficiently small has the effect of making the set over which we index in Theorem 3.17 more complicated. (3)Computations suggest that, in this case, the Newton and Hodge polygons associated to never touch (after the first vertex where they touch for trivial reasons) and therefore the methods of [10] cannot be used to describe the slopes.…”

“…In this case, weights have several components and therefore the boundary is more complicated as one can approach it via each of the components separately. Computations in [3] suggest that, again for weights near the boundary (see [3, Definition 2.1.8] for a precise definition), slopes behave in a highly predictable way but in general are not a union of arithmetic progressions with common difference. In this more general setting, one observes that the multiset of slopes appears to be completely determined by a simple recipe starting with slopes of classical Hilbert modular forms of some small weight.…”

$2$-adic slopes of Hilbert modular forms over $\mathbb{Q}(\sqrt{5})$

“…c This is due to the fact that the associated Newton and Hodge polygons in general only touch at the base point (see 3.18). 1 This theorem is in line with [Birb,Conjecture 4.7.9], appropriately generalized to this situation where the level is not sufficiently small. The proof is essentially an extended exercise on p-adic matrix analysis which we do not know how to generalize to many other situations.…”

The Jacquet–Langlands correspondence for overconvergent Hilbert modular forms

“…The weight space is a rigid analytic variety that allows us to make precise the idea of modular forms 'living' in p-adic families. Following [5], we begin with the classical definition of a weight of a Hilbert modular form.…”