Nonfreeness of some algebras of hermitian modular forms

TL;DR

This paper investigates the algebraic structure of hermitian automorphic forms associated with specific lattices and quadratic fields, demonstrating nonfreeness for certain discriminants and providing dimension estimates for potential freeness cases.

Contribution

It establishes that for discriminants greater than 7, the algebras of hermitian automorphic forms are not free, and offers dimension bounds for cases where freeness might occur.

Findings

Algebras are not free for d > 7

Dimension estimates for d=7 and d=3

Comparison with known results for d=3

Abstract

We study the algebras of hermitian automorphic forms for the lattice $L_n=diag(1,1,\ldots,1,-1)$ and for the field $K=\mathbb{Q}(\sqrt{-d})$ such that $p=2$ is unramified and the ring of integers $\mathcal{O}_K$ is a p.i.d. We prove that for $d>7$ these algebras can't be free. When $d=7$ and $d=3$ we give an estimate for the dimension of the symmetric spaces for which these algebras might be free. We also compare our results with the known results for $d=3$.