Computing Jacobi's $θ$ in quasi-linear time

Jacobi's $θ$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $θ(z,τ)$, for $z, τ$ 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 $θ$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $θ(z, τ)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $τ\in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $θ$.

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