Elliptic curve isogenies and their embedding into homomorphisms of p-adic tate modules

Abstract

This thesis investigates the interplay between the morphism spaces of elliptic curves and their associated Tate modules. Specifically, we focus on proving the injectivity of the map Hom (E1,E2) ⊗ Zp → Hom (Tp (E1) , Tp (E2)) , where E1 and E2 are elliptic curves, and Tp (E) denotes the p-adic Tate module associated with the elliptic curve E. The result connects the algebraic structure of morphisms over elliptic curves with the module-theoretic properties of their Tate modules. We employ tools from algebraic geometry and p-adic number theory, focusing on the role of endomorphism rings and Tate modules. This work contributes to understanding how information about elliptic curve morphisms is preserved and reflected in the realm of p-adic arithmetic.