Fast evaluation of Riemann theta functions in any dimension

Abstract

We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library and vastly outperforms existing software. As an application, we evaluate the theta constants attached to certain special abelian varieties of dimension 6 to construct explicit polynomials of degree 65 over $\mathbb{Q}$ with conjectural Galois group $\mathrm{SL}_2(\mathbb{F}_{64})$.