Section: New Results

Computing Theta Functions in Quasi-linear Time in Genus 2 and Above

Participants : Hugo Labrande, Emmanuel Thomé [contact] .

We study the multiprecision computation of the theta function in genus 2. We extend the quasi-linear algorithm for Jacobi's theta to genus 2, generalizing the approach we undertook in previous work; this required finding workarounds, most notably for the choice of signs and for being able to apply Newton's method. We also give an outline of an algorithm for the theta function in genus $g$, but the workarounds we found in genus 2 would need to be generalized to this case before claiming any sort of result in genus $g$ [6].

We released along with this work a Magma implementation of our fast genus 2 algorithm, along with an implementation of a somewhat naive (but previously state-of-the-art) algorithm for genus 2. Our results show that our algorithm is faster than the naive one for precisions greater than 3,000 digits.