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.
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$\begingroup$ Have you looked at this paper of Kestutis Cesnavicius: imo.universite-paris-saclay.fr/~kestutis.cesnavicius/…? I think one ought to be able to adapt the usual char 0 proof, which involves the fact that the group of cohomology classes that are unramified outside a finite set of primes is finite, to the positive char case by working with fppf cohomology. The main theorem in the linked paper has some extra assumptions, but I believe that the finiteness should be extractable in full generality. Have a look, in particular, at Section 5. $\endgroup$– Alex B.Commented Oct 26, 2022 at 12:49
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$\begingroup$ Sorry about the rushed comment! I am writing from memory. I can check this myself later and may leave a proper answer. $\endgroup$– Alex B.Commented Oct 26, 2022 at 12:55
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$\begingroup$ @Alex B. Thank you for this! I will have a look. I thought this was not a recent result, though. $\endgroup$– Damian RösslerCommented Oct 26, 2022 at 13:21
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$\begingroup$ Perhaps not directly relevant but I should also mention this paper of Brian Conrad on finiteness of Sha over global function fields: math.stanford.edu/~conrad/papers/cosetfinite.pdf $\endgroup$– David Benjamin LimCommented Oct 28, 2022 at 5:57
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$\begingroup$ @David Benjamin Lim. Thank you for this reference. I will have a look. $\endgroup$– Damian RösslerCommented Oct 28, 2022 at 8:28
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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.
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$\begingroup$ Great, thank you so much! This is indeed what I was looking for. $\endgroup$ Commented Oct 26, 2022 at 14:19
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1$\begingroup$ The case when the characteristic of the field is coprime to the degree of the isogeny is of course older, like Theorem 5 in Lang-Tate. (jstor.org/stable/2372778) $\endgroup$ Commented Oct 26, 2022 at 14:21
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$\begingroup$ @Chris Wuthrich. Yes of course. My question was really for the inseparable case. $\endgroup$ Commented Oct 26, 2022 at 16:15