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

TL;DR

This paper explicitly determines the cones of weights for sections of automorphic line bundles on Goren-Oort strata in Hilbert modular varieties, revealing their structure and minimal cones, with implications for eigenforms and Hecke eigenvalues.

Contribution

It explicitly characterizes the cones of weights and minimal cones for all Goren-Oort strata in Hilbert modular varieties, and relates these to eigenforms and Hecke eigenvalues.

Findings

Cones of weights are explicitly determined for all strata.

Minimal cones are explicitly identified for each stratum.

Existence of eigenforms with weights in minimal cones sharing Hecke eigenvalues.

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 $\kappa$ can be defined as a section of an automorphic line bundle $\mathcal{L}_\kappa$ on $X$. We consider sections of $\mathcal{L}_\kappa$ (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 $X_T$.