In Chapter III, Theorem 7.4 of *The Arithmetic of Elliptic Curves* (first edition), Silverman gives the following lemma and proof:

Lemma: Let $M \subset Hom(E_1, E_2)$ be a finitely generated subgroup, and let $M^{div} = \{ \phi \in Hom(E_1, E_2) : [m] \circ \phi \in M$ for some integer $m \geq 1\}$. Then $M^{div}$ is also finitely generated.

Proof:Extend the degree mapping to the finite dimensional real vector space $M \otimes \mathbb{R}$, which we equip with the natural topology inherited from $\mathbb{R}$. Then the degree mapping is clearly continuous, so the set $U = \{\phi \in M \otimes \mathbb{R} : deg(\phi) < 1 \}$ is an open neighborhood of $0$. Further, since $Hom(E_1, E_2)$ is a torsion-free $\mathbb{Z}$-module, there is a natural inclusion $$M^{div} \subset M \otimes \mathbb{R};$$ and clearly $$M^{div} \cap U = \{ 0 \},$$ since every non-zero isogeny has degree at least 1. Hence $M^{div}$ is a discrete subgroup of the finite dimensional vector space $M \otimes \mathbb{R}$, so it is finitely generated.

When I first saw it, this proof felt like absolute voodoo to me. Tensor an endomorphism ring with the reals? What, so you can take $\pi$ or $e$ times an endomorphism?

The point is clearer to me now -- real vector spaces are a nice place to argue that things are finitely generated -- but I'm not sure I would have seen to do this if the proof were left as an exercise.

Does this idea have any natural context? Are proofs like this commonplace in some other area of math?

Thank you!

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