On the geometry of the Humbert surface of square discriminant

For every positive integer N we determine the Enriques--Kodaira type of the Humbert surface of discriminant N² 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 N² 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 GL₂(Z/NZ) and which may be of independent interest.