Geometric construction of modular polynomials with level structures

TL;DR

This paper introduces an algebraic approach to constructing modular polynomials for higher level invariants of elliptic curves, including Montgomery and Hessian models, with proven properties and a new computation algorithm.

Contribution

It provides a novel algebraic method for constructing higher level modular polynomials directly related to elliptic curve models, with proofs of their properties and an explicit computation algorithm.

Findings

Existence of modular polynomials with integer coefficients

Symmetry and irreducibility in certain cases

An algorithm based on deformation methods for computation

Abstract

The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on elliptic curves. In addition, computing the modular polynomial itself is also an important problem, and various algorithms to compute it have been proposed. On the other hand, modular polynomials for other invariants of higher level structures have also been studied. For example, the modular polynomials for the Legendre $\lambda$-invariant and the Weber functions are well-known. In this paper, we give another approach to construct modular polynomials of higher level purely algebraically. In particular, we show the existence of modular polynomials for invariants directly related to models of elliptic curves, such as the coefficients of Montgomery and Hessian curves. We also show that these modular polynomials have integer coefficients and are symmetric and irreducible in certain cases, and give an algorithm to compute them, which is based on the deformation method by Kunzweiler and Robert.