Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 32 / 2 / i / Solution

An isogeny of elliptic curves over $k$ is a nonconstant $k$-morphism $\phi :E_1\to E_2$ which sends $O_1$ to $O_2$. A nonconstant morphism of smooth projective curves is finite and surjective. Its compatibility with the elliptic curve group law from Riemann-Roch can be seen without assuming it in the definition.

Identify a point $P$ with the degree-zero divisor class $[P-O_1]$. A finite morphism pushes forward divisors, and pushes a principal divisor to the principal divisor of the norm of its rational function. It therefore induces a group homomorphism on degree-zero Picard groups of a curve. On geometric points, the pushforward of $[P-O_1]$ is $[\phi (P)-O_2]$. Since point addition is addition of these divisor classes, this provesThe equality is defined over $k$, so in particular it holds on the groups of $k$-rational points and after every field extension. The argument also covers inseparable isogenies of elliptic curves; separability is not part of the definition.