A polynomial formula for the Albanese map of the Cartwright-Steger surface

Source abstract

The Albanese fibration of the Cartwright--Steger surface is a genus-$19$ Lefschetz fibration over a torus. We give an explicit formula for it: in the projective model of Borisov and Yeung, the map is represented by three homogeneous polynomials of degree $13$ with integer coefficients, and its target is the elliptic curve $Y^2Z=X^3-3888Z^3$. A recognition theorem shows that it suffices to verify one polynomial identity on the surface. A coefficient bound allows us to establish the identity over $Q$ from computations modulo primes. We also determine exact coordinates for the three nodes and their images on the elliptic curve.