Partial Hasse invariants for genus zero curves in Hilbert modular varieties
Gabriele Bogo, Yingkun Li
TL;DR
The paper addresses lifting partial Hasse invariants for genus-zero curves embedded in Hilbert modular varieties by connecting characteristic-$p$ geometry to twisted modular forms via Picard–Fuchs equations along affine Kobayashi geodesics. It constructs explicit lifts $h_{p,j}$ of the partial Hasse invariants with controlled weights and divisors, proving that their $t$-expansions are constant modulo $p$ and relating the zeros to the non-ordinary locus; in genus zero, the degree of the associated polynomials $\mathrm{ph}_{p,j}(t)$ is tied to the dimensions of twisted modular form spaces, giving Deuring-like counts for Teichmüller curves. The results include minimal-weight lifts under elliptic-point assumptions and explicit computations for Teichmüller curves $W_D$ in Hilbert modular surfaces and for the triangle curve $W_5$, with representations in terms of hypergeometric functions. A constructive model-based method is provided to express partial Hasse invariants as polynomials in modular forms, enabling explicit $t$-expansions and Hasse–Witt matrix descriptions in worked examples.
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 Teichmüller 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.