A computational proof of the existence of the Dual Isogeny

TL;DR

This paper provides a computational proof for the existence of the dual isogeny of an elliptic curve, utilizing Vélú's formulas rather than traditional Galois correspondence methods.

Contribution

It introduces a novel computational approach to prove the dual isogeny existence, offering an alternative to classical algebraic methods.

Findings

Computational proof confirms dual isogeny existence for elliptic curves.

Uses Vélú's formulas as the main computational tool.

Provides an alternative to Galois correspondence in isogeny proofs.

Abstract

For $E$ an elliptic curve over a perfect field $K$, we present a proof of the existence of the dual isogeny $\hat{\phi}$ using computational methods linked to V\'elu's formulae instead of the standard Galois correspondence method.