Wang–Williams conjecture on free algebras of unitary modular forms

Let $E$ be an imaginary quadratic field with odd discriminant $-D$, let $(L,\langle\phantom{x},\phantom{x}\rangle)$ be a Hermitian lattice over $\mathcal{O}_E$ of signature $(1,n)$ for $n>2$, and let $\Gamma<\operatorname{U}(L)$ be an arithmetic subgroup. Write $M_*(\Gamma)$ for the graded algebra of modular forms for $\Gamma$. Wang–Williams' conjecture. The graded algebra $M_*(\Gamma)$ is not a free algebra for $n>5$. This conjecture concerns the classification of free graded algebras of modular forms on unitary groups; the paper proves non-freeness in higher dimensions and therefore provides partial evidence, while the stated range $n>5$ remains unresolved here.