On the space of slow growing weak Jacobi forms (Q2161356)

Weak Jacobi forms of slow growth are weak Jacobi forms of weight zero for which certain sums of Fourier coefficients satisfy a surprising amount of cancellation. Slow growing weak Jacobi forms are interesting because they lift to Borcherds products that have moderate growth when expanded about Humbert surfaces in the Siegel upper half-space. This paper studies the spaces of slow growing weak Jacobi forms and proves that there exist nonzero examples of all indices up to \(m = 1000\). In addition, the authors conjecture a test for slow growth in terms of the coefficients of a certain vector-valued modular function.