A quasi-linear time algorithm for computing modular polynomials in dimension 2

TL;DR

This paper introduces a quasi-linear time algorithm for computing modular polynomials in dimension 2 using invariants from theta constants, with theoretical analysis and experimental validation.

Contribution

It generalizes Dupont's method to new invariants and proves quasi-linear complexity under heuristics.

Findings

Algorithm is quasi-linear in output size.

Properties of modular polynomials from theta quotients are analyzed.

Experimental results support theoretical claims.

Abstract

We propose to generalize the work of R\'egis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove under some heuristics that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.