Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated

TL;DR

This paper proves that the ring of Siegel-Jacobi forms with fixed degree and bounded weight-to-index ratio is not finitely generated, using toroidal b-divisors and convex geometry, and confirms related conjectures and formulas.

Contribution

It demonstrates the non-finite generation of certain Siegel-Jacobi form rings and proves a conjecture about their representation as line bundle sections.

Findings

Ring of Siegel-Jacobi forms is not finitely generated.

Confirmed Kramer’s conjecture on form representation.

Reproduced Tai’s asymptotic dimension formula.

Abstract

We show that the ring of Siegel-Jacobi forms of fixed degree and of fixed or bounded ratio between weight and index is not finitely generated. Our main tool is the theory of toroidal b-divisors and their relation to convex geometry. As a byproduct of our methods, we prove a conjecture of Kramer about the representation of all Siegel-Jacobi forms as sections of certain line bundles and we recover a formula due to Tai for the asymptotic dimension of the space of Siegel-Jacobi forms of given ratio between weight and index.