In the article ‘Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated’, Ana María Botero (University of Regensburg, Germany), José Ignacio Burgos Gil (ICMAT-CSIC) and David Holmes and Robin de Jong (both from the Mathematical Institute of Leiden University, The Netherlands), (Mathematical Institute of Leiden University, The Netherlands) prove what the title indicates: Siegel-Jacobi shape rings of bounded relative index are not finitely generated, contrary to previous beliefs.

In science, no law is immutable. All laws must be continuously tested against reality, and if an experiment shows that a law is inaccurate, it is replaced by another. In mathematics, however, we have proofs. If a result is proven, it is considered set in stone and immutable. However, the complexity of modern mathematics often makes it impossible to rigorously verify all results during peer review, and errors can appear in various publications. These errors are typically not discovered by reading the proofs, but because their consequences contradict other established results.

The origin of this article is a result published more than 30 years ago that seemed too good to be true. Siegel–Jacobi modular forms are a generalization of theta functions and are of great importance in the study of abelian varieties and families of abelian varieties. Just as classical modular forms are classified by a single number, the weight, Siegel–Jacobi forms are classified by two numbers: the weight and the index. One of the classical problems in the theory is the calculation of the dimension of the space of Siegel–Jacobi forms with a given weight and index. A result by Runge, published in the 1990s, states that the ring of Siegel–Jacobi forms with a fixed ratio between weight and index is finitely generated. This would imply that the dimensions of Siegel–Jacobi forms should be relatively easy to calculate—somewhat like counting the number of lattice points contained within the multiples of a polyhedron with rational vertices.

In contrast, the problem of determining the dimension of the Siegel–Jacobi forms is more complex and resembles counting the number of lattice points within the multiples of a curved convex shape.

In this article, the authors use the theory of b-divisors on toroidal varieties to prove that the ring of Siegel–Jacobi forms with a fixed and positive ratio between weight and index is never finitely generated. As an application of these techniques, they prove a conjecture of Kramer regarding the interpretation of Siegel–Jacobi forms as global sections of certain fiber bundles and provide a new proof of a formula by Tai concerning the asymptotic behavior of the dimension of the Siegel–Jacobi forms.

Reference: Ana María Botero, José Ignacio Burgos Gil, David Holmes, Robin de Jong. “Rings of Siegel–Jacobi forms of bounded relative index are not finitely generated,” Duke Mathematical Journal, Duke Math. J. 173(12), 2315-2396, (1 September 2024).