Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $\phi$ is the well-known Selmer group ${\rm Sel}(\phi)\subseteq H^1(K,{\rm ker}(\phi))$, which contains $B(K)/\phi(A(K))$. It is classical that this group is finite but I couldn't find a reference in the situation where $K$ has positive characteristic. I have it in my memory that J. Milne proved this in the early 1970s but I wasn't able to find the corresponding article. I would be grateful for any help.



Reference request. Finiteness of the Selmer group
The paper is Milne, J. S. Elements of order p in the Tate-Šafarevič group. Bull. London Math. Soc. 2 (1970), 293–296. He deduces his statement about the Tate-Shafarevich group from a statement about the Selmer group. He also notes that if you omit any places in the definition of these groups, then you may get infinitely many elements of order p.


