Analytic continuation on Shimura varieties with μ-ordinary locus

Abstract

Abstract. We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the µ-ordinary locus (with an explicit bound). We then define the overconvergent modular forms (of classical weight), as well as the relevant Hecke operators. Finally, we show how an analytic continuation argument can be adapted to this case to prove a classicality theorem, namely that an overconvergent modular form which is an eigenform for the Hecke operators is classical under certain assumptions.

Families of Picard modular forms and an application to the Bloch–Kato conjecture

“…Define the moduli space over which parametrizes data where and is a totally isotropic -module for (i.e. ) of -height (and thus ) such that As remarked by Bijakowski in [Bij16], the second condition is implied by the first one and the isotropy condition. We then define two projections, where is the forgetful map which sends to and sends to .…”

A trace formula approach to control theorems for overconvergent automorphic forms

“…After the first version of this paper this paper was written we learned that Goldring and Nicole ([GN]) has proved the affineness of the µ-ordinary locus for general unitary Shimura varieties at a prime of good reduction. In particular, this means that one can run the arguments of this paper in the case when F is CM with p unramified to obtain control theorems for Shimura varieties with vanishing ordinary locus, as long as one can extend Hida's calculations for integrality of Hecke operators at p. These calculations were done in [Bij3], which proves control theorems in at Iwahori level in many cases with empty ordinary locus. With this, the arguments of this paper generalise to give higher level control theorems as well.…”

“…Control theorems for the Shimura varieties considered in this paper have been proved previously in the Iwahori case using the analytic continuation method by Pilloni and Stroh ([PS2]) and later vastly generalized and extended to include higher level cases by Bijakowski in a series of papers ([Bij1,Bij2,Bij3]). We remark that the only reason for restricting our attention in this paper to the special case when F is an imaginary quadratic field (in the context of the Shimura varieties considered in [HT], as opposed to F being a CM field containing an imaginary quadratic field) was the need for the µ-ordinary locus to be affine.…”

Duality, refined partial Hasse invariants and the canonical filtration

“…Indeed, one can show that on strict neighborhoods of the µ-ordinary locus in some Shimura varieties, canonical filtrations (with my definition) always exist. Note that the understanding of such neighborhoods would be a key step in constructing overconvergent modular forms of any weight for Shimura varieties with empty ordinary locus (see [Bi2] for the definition of such overconvergent modular forms of classical weight).…”