Geometric construction of modular polynomials with level structures
Hiroshi Onuki, Yukihiro Uchida, Ryo Yoshizumi
TL;DR
The paper develops a purely algebraic framework for constructing modular polynomials of higher level for invariants tied to elliptic curve models, notably Montgomery and Hessian forms. It introduces invariants with good models, proves existence, integrality, and symmetry properties of the corresponding modular polynomials $Φ_m^α$, and provides a CRT-based deformation algorithm (extending Kunzweiler–Robert) to compute them efficiently. The authors validate the approach with explicit Montgomery and Hessian examples, prove irreducibility under computability assumptions, and present experimental results that support the theory and demonstrate practical computation up to small primes. This work broadens the scope of modular-polynomial constructions beyond classical $j$-invariants, enabling higher-level isogeny-based computations for a wider class of elliptic-curve models.
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 $λ$-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.