Hilbert surfaces, modular forms, and Siegel-Veech constants

TL;DR

This paper computes Siegel-Veech constants for saddle connections on translation surfaces in Prym eigenform loci, revealing they are identical across these loci, and relates Euler characteristics of certain Hilbert modular surfaces.

Contribution

It provides explicit values of Siegel-Veech constants for Prym eigenform loci and uncovers a surprising uniformity among them, also relating Euler characteristics of associated modular surfaces.

Findings

Siegel-Veech constants are the same for all Prym eigenform loci in the specified moduli space.

Euler characteristic of Hilbert modular surfaces relates simply to that of their product locus.

The phenomenon observed extends known results for principally polarized Abelian surfaces.

Abstract

We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $\Omega \mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with $(1,2)$-polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge.