Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weber modular curves and modular isogenies

Leonardo Colò, David Kohel

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.

Weber modular curves and modular isogenies

Abstract

We study the modular curves defined by Weber functions, and associated modular polynomials, action of , 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.

Paper Structure

This paper contains 5 sections, 13 theorems, 54 equations, 1 table.

Table of Contents

  1. Introduction
  2. Weber functions and modular polynomials
  3. Weber curves
  4. Weber supersingular modules
  5. Conclusion

Key Result

Proposition 1

The modular polynomial $\Phi_\ell(x,y)$ of prime level $\ell$ with respect to the Weber function satisfies the transformation: with respect to a primitive $24$-th root of unity $\zeta_{24}$.

Theorems & Definitions (21)

  • Proposition 1
  • Corollary 2
  • Proposition 3
  • proof
  • Corollary 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 11 more