Partial Hasse invariants for genus zero curves in Hilbert modular varieties

TL;DR

This paper constructs characteristic-zero lifts of partial Hasse invariants for genus zero curves in Hilbert modular varieties, linking the non-ordinary locus size to modular form spaces and providing explicit computations for Teichmüller curves.

Contribution

It introduces a new method to lift partial Hasse invariants using Picard-Fuchs equations and relates these to the structure of modular forms on genus zero curves.

Findings

Explicit formulas for the non-ordinary locus size on Teichmüller curves

Deuring-like formulas derived for specific cases

Insights into the reduction of modular forms on Fuchsian groups

Abstract

We construct characteristic-zero lifts of partial Hasse invariants for genus zero non-compact curves in Hilbert modular varieties. The construction is based on recent results on the associated Picard-Fuchs differential equations. As an application, we relate the size of the non-ordinary locus of the modulo $p$ reduction of these curves to the dimension of spaces of (twisted) modular forms. We compute it explicitly for several Teichm\"uller curves, obtaining Deuring-like formulae. Moreover, we study the modulo $p$ reduction of (twisted) modular forms on not necessarily arithmetic genus-zero Fuchsian groups with modular embedding.