On the geometry of the Humbert surface of square discriminant

TL;DR

This paper classifies the geometric types of Humbert surfaces of square discriminant, focusing on their fixed point loci under involutions and introducing divisors with genus formulas relevant to modular curves and abelian surfaces.

Contribution

It determines the Enriques--Kodaira types of Humbert surfaces of discriminant N^2 and constructs fixed divisors with genus formulas, advancing understanding of abelian surface moduli.

Findings

Classification of Humbert surface types for discriminant N^2

Construction of fixed point divisors under involution

Genus formulas for these divisors, including modular curves

Abstract

For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key step in the proof is to analyse the fixed point locus of a Fricke-like involution on the Hilbert modular surface of discriminant $N^2$ which was studied by Hermann and by Kani and Schanz. To this end, we construct certain "diagonal" Hirzebruch--Zagier divisors which are fixed by this involution. In our analysis we obtain a genus formula for these divisors, which includes the case of modular curves associated to (any) extended Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ and which may be of independent interest.