Cones of Weights and Minimal Cones of the Goren-Oort Strata in Hilbert modular varieties

TL;DR

This work determines the cones of weights for all Goren-Oort strata in Hilbert modular varieties, showing that, unlike the ambient case, their cones are not generally generated solely by partial Hasse invariants. It proves that every cone is generated by pullbacks of Hasse invariants from related strata and gives an explicit description of the minimal cone ${\mathcal C}^{\min}_T$ for each stratum $X_T$, including a precise description in the inert- and tilde-parameter cases. For Shimura varieties attached to ${\operatorname{Res}}_{F/\mathbb{Q}} {\operatorname{GL}}_2$, the authors show that any nonzero prime-to-$p$ Hecke eigenform on a stratum has a companion eigenform with the same eigenvalues whose weight lies in the minimal cone, highlighting a refinement of weight distribution under Hecke action. The approach combines constructions of GO strata, explicit cone generators, stratifications, and descent to the GL$_2$ setting, offering a strategy to compute minimal weights on more general Shimura varieties. The results advance understanding of Serre-type weight conjectures and the relation between automorphic forms and geometric stratifications in characteristic $p$.

Abstract

Let $p$ be a prime, $F$ a totally real field in which $p$ is unramified, and $X/\overline{\mathbb{F}}_p$ a Shimura variety associated to ${\rm Res}_{F/\mathbb{Q}} {\rm GL}_2$ (or a PEL Hilbert modular variety). A mod $p$ Hilbert modular form of weight $κ$ can be defined as a section of an automorphic line bundle $\mathcal{L}_κ$ on $X$. We consider sections of $\mathcal{L}_κ$ (forms) over a Goren-Oort stratum $X_T$ inside $X$, and define the cone of weights of $X_T$ to be the $\mathbb{Q}^{\geq 0}$-cone generated by the weights of all nonzero forms on $X_T$. We explicitly determine the cone of weights of all strata, showing in particular that they are not in general generated by the weights of the associated Hasse invariants. Using this, we define a notion of minimal cone for each stratum, and explicitly determine the minimal cones of all strata. When $X$ is a Shimura variety associated to ${\rm Res}_{F/\mathbb{Q}} {\rm GL}_2$, we prove that for every nonzero eigenform $f$ for the prime-to-$p$ Hecke algebra on a stratum $X_T$, there is another eigenform with the same Hecke eigenvalues which has weight in the minimal cone of $X_T$.

Theorems & Definitions (67)

• Remark 2.1
• Definition 2.2
• Corollary 2.3
• Definition 3.1
• Lemma 3.2
• proof
• Proposition 4.1
• proof
• Lemma 4.2
• proof