Weber modular curves and modular isogenies

TL;DR

This paper investigates Weber modular curves and their associated isogeny graphs, focusing on supersingular elliptic curves, with applications to cryptography and Galois representations.

Contribution

It introduces new insights into Weber modular curves, their isogeny graphs, and applications to cryptography and explicit Galois representations.

Findings

Analysis of Weber functions and modular polynomials

Characterization of isogeny graphs for supersingular elliptic curves

Applications to efficient cryptographic isogeny computations

Abstract

We study the modular curves defined by Weber functions, and associated modular polynomials, action of $\mathrm{SL}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine, particularly for supersingular elliptic curves. In addition to applications to efficient isogeny computation in cryptographic applications, we present an application to explicit Galois representations.