4.4 Article

Reflective Modular Forms: A Jacobi Forms Approach

期刊

INTERNATIONAL MATHEMATICS RESEARCH NOTICES
卷 2021, 期 3, 页码 2081-2107

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OXFORD UNIV PRESS
DOI: 10.1093/imrn/rnz070

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  1. Labex Centre Europeen pour les Mathematiques, la Physique et leurs interactions in the University of Lille [ANR-11-LABX-0007-01]

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The paper provides an explicit formula to express the weight of 2-reflective modular forms and proves the non-existence of 2-reflective lattices of signature (2, n) when n is greater than or equal to 15 and not equal to 19. Applications of the results include a simple proof of Looijenga's theorem and classification of reflective modular forms on lattices of large rank.
We give an explicit formula to express the weight of 2-reflective modular forms. We prove that there is no 2-reflective lattice of signature (2, n) when n >= 15 and n not equal 19 except the even unimodular lattices of signature (2, 18) and (2, 26). As applications, we give a simple proof of Looijenga's theorem that the lattice 2U circle plus 2E(8)(-1) circle plus <-2n > is not 2-reflective if n > 1. We also classify reflective modular forms on lattices of large rank and the modular forms with the simplest reflective divisors.

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