Computing Jacobi's $\theta$ in quasi-linear time

TL;DR

The paper presents a quasi-linear time algorithm for computing Jacobi's theta function with high precision, significantly improving over previous methods by leveraging properties of theta-constants and quasi-periodicity.

Contribution

It introduces a novel algorithm that computes Jacobi's theta function in asymptotically faster time, reducing complexity from sub-quadratic to quasi-linear in precision.

Findings

Computes theta-constants in asymptotically faster time

Achieves theta function computation in O(M(P) log P) bit operations

Applicable for any tau in fundamental domain and reduced z

Abstract

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $\theta$.