The Geometry of Drinfeld Modular Forms

2023/10/30 by Jesse Franklin, Franklin, Jesse
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.19623

openalex publication_date 2023/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups Γ≤ \GL₂(\bbF_q[T]). In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for Γ₂ and the algebra of Drinfeld modular forms for Γ₂, where Γ₂ is the subgroup of square-determinant matrices in Γ. This allows one to compute the latter ring by geometric invariants using the techniques of Voight, Zureick-Brown and O'Dorney. We also show how to decompose the algebra of modular forms for Γ₂ into a direct sum of two algebras of modular forms for Γ and generalize this result to a larger class of congruence subgroups.

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