Timeline for Selmer of an abelian variety versus that of its dual.

9 events

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Mar 18, 2011 at 12:27	history	edited	jvo		edited tags; edited tags
Mar 10, 2011 at 7:13	vote	accept	jvo
Mar 10, 2011 at 7:12	comment	added	jvo		Ah yes, thanks for pointing this out!
Mar 10, 2011 at 0:47	comment	added	Chris Wuthrich		Did you really mean to define $\mathfrac{S}(A/K)$ the way you did ? The compact Selmer group, usually denoted with $\mathfrak{S}(A/K)$, is in the Galois cohomology of $T_p A$, not $A_{p^{\infty}}$. If you change this in the definition then your remarks afterwards are correct. If you want to keep this definition, then the answer of Remke Kloosterman tells you what should be true.
Mar 9, 2011 at 20:46	answer	added	Remke Kloosterman		timeline score: 5
Mar 9, 2011 at 17:28	comment	added	jvo		I think that any isomorphism should do. And I agree that if p divides the isogeny, then it is not at all clear whether or not such an isomorphism should hold!
Mar 9, 2011 at 17:24	history	edited	jvo	CC BY-SA 2.5	added 4 characters in body
Mar 9, 2011 at 14:50	comment	added	Chris Wuthrich		Is the last $\cong$ in your question, just any isomorphism of groups or do you want to take the one induced by an isogeny from $A$ to its dual ? If $p$ divides the degree of this isogeny then it might be that only in one direction the isogeny gives an isomorphism (and maybe none, I don't know), but I have no examples at hand.
Mar 9, 2011 at 14:18	history	asked	jvo	CC BY-SA 2.5