{
  "id": "2105.14892",
  "version": "v1",
  "published": "2021-05-31T11:32:17.000Z",
  "updated": "2021-05-31T11:32:17.000Z",
  "title": "Free algebras of modular forms on ball quotients",
  "authors": [
    "Haowu Wang",
    "Brandon Williams"
  ],
  "comment": "42 pages, comments are welcome",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In this paper we study algebras of modular forms on unitary groups of signature $(n,1)$. We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary we obtain a criterion that guarantees in many cases that, if $L$ is an even lattice with complex multiplication and the ring of modular forms for its orthogonal group is a polynomial algebra, then the ring of modular forms for its unitary group is also a polynomial algebra. We prove that a number of rings of unitary modular forms are freely generated by applying these criteria to Hermitian lattices over the rings of integers of $\\mathbb{Q}(\\sqrt{d})$ for $d=-1,-2,-3$. As a byproduct, our modular groups provide many explicit examples of finite-covolume reflection groups acting on complex hyperbolic space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-31T11:32:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F55",
      "11E39",
      "14G35"
    ],
    "keywords": [
      "free algebras",
      "ball quotients",
      "unitary modular forms",
      "unitary group",
      "polynomial algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}