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<p>In this paper, we study algebras of modular forms on unitary groups of signature (<i>n</i>,&#xa0;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 <i>L</i> 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2025_538_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Q}(\sqrt{d})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>d</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2025_538_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(d=-1,-2,-3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. As a byproduct, our modular groups provide many explicit examples of finite-covolume reflection groups acting on complex hyperbolic space.</p>

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Free algebras of modular forms on ball quotients

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月29日

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  • Haowu Wang,
  • Brandon Williams

摘要

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 \(\mathbb {Q}(\sqrt{d})\) Q ( d ) for \(d=-1,-2,-3\) d = - 1 , - 2 , - 3 . As a byproduct, our modular groups provide many explicit examples of finite-covolume reflection groups acting on complex hyperbolic space.