Computing theta functions in quasi-linear time in genus two and above

Abstract

We outline an algorithm to compute $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus two in quasi-linear time, borrowing ideas from the algorithm for theta constants and the one for $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus one. Our implementation shows a large speed-up for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus $g$.