# Nonfreeness of algebras of symmetric Hilbert modular forms of even weight for $\mathbb{Q}(\sqrt{d})$ where d>5

## TL;DR

This paper investigates the structure of algebras of symmetric Hilbert modular forms for real quadratic fields, demonstrating that they are not free for most cases by comparing geometric volumes.

## Contribution

It proves that the algebras of symmetric Hilbert modular forms are nonfree for all real quadratic fields except for a few small discriminants.

## Key findings

- Algebras are nonfree for all d > 5 except d=2,3,5
- Volume comparisons are used to establish nonfreeness
- Results extend understanding of modular form algebra structures

## Abstract

We study the algebras of symmetric Hilbert modular forms of even weight for $\mathbb{Q}(\sqrt{d})$, considering them as modular forms for the orthogonal group of the lattice with signature (2,2). Comparing the volume of the corresponding symmetric domain with the volume of the Jacobian of the generators of these algebras, we prove that for all d, except for d=2, 3, 5 these algebras can't be free.