Appendix to a paper [arXiv:1809.08623] of B. Williams

TL;DR

This appendix provides explicit birational equivalences between models of Hilbert modular surfaces for specific quadratic fields, connecting recent models with those previously established by Elkies and Kumar.

Contribution

It establishes explicit birational maps linking new models of Hilbert modular surfaces to classical ones, enhancing understanding of their geometric relationships.

Findings

Explicit birational equivalences for ${ m Q}( oot 2 29)$ and ${ m Q}( oot 2 37)$ models

Connections between recent and classical Hilbert modular surface models

Improved understanding of the geometric structure of these surfaces

Abstract

In this appendix to a paper [arXiv:1809.08623] by B. Williams, we give birational equivalences between the models of the Hilbert modular surfaces for ${\mathbb Q}(\sqrt{29})$ and ${\mathbb Q}(\sqrt{37})$ given there and those previously found by Elkies and Kumar.