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A polynomial formula for the Albanese map of the Cartwright-Steger surface
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 Y2Z=X3−3888Z3. 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.
Related
- The Real Albanese Map and Smooth Topology of the Cartwright--Steger Quotient
- The real quotient of the Cartwright-Steger surface
- Finite Cyclic Fundamental Groups and Exotic Universal Covers from the Cartwright--Steger Surface
- Torus Surgery Constraints and Intermediate Covers of the Cartwright--Steger Surface
- A monodromy relation for the Cartwright-Steger fibration
BibTeX
@misc{szabo2026polynomial,
title = {{A polynomial formula for the Albanese map of the Cartwright-Steger surface}},
author = {Zoltán Szabó},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23124},
eprint = {2609.23124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23124},
}
RIS
TY - UNPB
TI - A polynomial formula for the Albanese map of the Cartwright-Steger surface
AU - Zoltán Szabó
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23124
UR - https://arxiv.org/abs/2609.23124
AB - 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.
ER -