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!



Proving finite generation by tensoring with $\mathbb{R}$
How do you prove that the ring of integers in a number field is finitely generated? One usually embeds them in $\mathbb{R}^{r\_1}\times\mathbb{C}^{r\_2}$. How do you prove that the units in a number field are finitely generated? One generally embeds them in a hyperplane in $\mathbb{R}^{r\_1+r\_2-1}$. Then one shows that the group sits as a discrete subgroup, hence is finitely generated. Further, by looking at the co-volume of the resulting lattice, one obtains important arithmetic invariants, namely the discriminant and the regulator. Anyway, one can call almost any technique a trick, but the idea of embedding a group into a real or complex vector space and using volume estimates to prove discreteness is quite a well-established technique. And if one simply has the group and a positive definnite quadratic form, as in the case of End$(E)$ or the canonical height on $E(\mathbb{Q})$, then it is very natural to tensor with $\mathbb{R}$ and extend the quadratic form to put a Euclidean structure on the resulting vector space. 


Of course, as Will Sawin says, one can alternatively view the embedding as being into the dual. But for example, using the height pairing on $E(\mathbb{Q})/tors$, the values of $\hat h$ are real, so even though $E(\mathbb{Q})/tors$ is a torsion-free $\mathbb{Z}$-module, the dual space that one uses has to be Hom$(E(\mathbb{Q})/tors,\mathbb{R})$.


I don't know that this really answers the question, but I hope it gives some indication of why it's not so unusual to think of embedding $M$ into a real vector space as the first step in proving that it is finitely generated. 


Finally, I should mention that I first saw the argument for proving that End$(E)$ is finitely generated in Mumford's *Abelian Varieties*, but I don't know the origins of the idea.


