Modular Polynomials

Wednesday, 15 February 2023

Abstract

Consider the classical modular $j$-function on the upper half-plane, and let $N$ be a positive integer. Then the modular polynomial $\Phi_N(X,j)$ is the minimal polynomial of $j(N\tau)$ over the field of modular functions $\mathbb{C}(j(\tau))$.

These polynomials, which are used in certain cryptosystems, have integer coefficients which are surprisingly large, yet many algorithms for computing them require an explicit upper bound on these coefficients.

I will outline joint work with Fabien Pazuki obtaining an explicit upper bound on the size of these coefficients.