2023/01/30 by Haowu Wang, Wang, Haowu
Mathematics · #11F55 #14B22 #17B67 #51F15 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2301.12606
openalex publication_date 2023/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A modular form on an even lattice M of signature (l,2) is called reflective if it vanishes only on quadratic divisors orthogonal to roots of M. In this paper we show that every reflective modular form on a lattice of type 2U⊕ L induces a root system satisfying certain constrains. As applications, (1) we prove that there is no lattice of signature (21,2) with a reflective modular form and that 2U⊕ D20 is the unique lattice of signature (22,2) and type U⊕ K which has a reflective Borcherds product; (2) we give an automorphic proof of Shvartsman and Vinberg's theorem, asserting that the algebra of modular forms for an arithmetic subgroup of O(l,2) is never freely generated when l≥ 11. We also prove several results on the finiteness of lattices with reflective modular forms.