The rings of Hilbert modular forms for $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$

TL;DR

This paper computes the generators and relations of the graded rings of Hilbert modular forms for the fields Q(√29) and Q(√37) using Borcherds products, overcoming previous obstructions in such calculations.

Contribution

It provides the first explicit descriptions of these graded rings for the specified fields, advancing understanding of Hilbert modular forms and Borcherds products.

Findings

Determined generators and relations for the graded rings.

Applied Borcherds products to specific Hilbert modular forms.

Overcame obstructions to compute these rings explicitly.

Abstract

We use Borcherds products and their restrictions to Hirzebruch-Zagier curves to determine generators and relations for the graded rings of Hilbert modular forms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. These seem to be the first cases where the graded ring can be computed despite obstructions to the existence of Borcherds products with arbitrary divisors.