ABSTRACT
Isogenies, which are homomorphisms between elliptic curves, play a fundamental role in the theory of elliptic curves since they allow us to relate one elliptic curve to another. In the first section, we describe the analytic theory over the complex numbers. In subsequent sections, we obtain similar results in the algebraic setting. Finally, we sketch how isogenies can be used to count points on elliptic curves over finite fields.
