Talk:Weil–Châtelet group

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The WC group is not the same thing as the Selmer group a far as I can tell from all of the literature I've ever read (e.g., Silverman's "Arithmetic of Elliptic Curves"). In fact, the Selmer group, unlike the Tate-Shafarevich group, doesn't even sit inside WC--instead, it's the kernel of H^1(A[f]) to all local WC's, f:A->A' some isogeny. 72.68.33.194 00:12, 24 April 2007 (UTC)


 * Indeed the Selmer group is not the same as the WC group (though it does sit inside it). I have just made that change, as well as some other additions. I think that the Tate-Shafarevich and Selmer groups are important enough to have their own articles, though right now this combined article is really no more than a stub. Explaining the relevance of Selmer groups to Iwasawa theory and of both the Tate-Shafarevich group and the Selmer group to BSD would be good. Also, does anyone know how to make the cyrillic letter Sha in math, I tried \Sha, but it did not work. RobHar 21:08, 3 August 2007 (UTC)

No, the Selmer group is not by definition a subgroup of the Weil-Chatelet group. The Selmer group is a certain subgroup of the Galois cohomology group with coefficients in the kernel of the isogeny f, i.e., with finite coefficients. This group surjects onto the Weil-Chatelet group, and the Selmer group can be defined as the complete preimage of the Tate-Shafarevich group under this morphism.

Also, I for one am interested in WC-groups over fields other than local and global fields, e.g. a finitely generated field. Better to write the statement positively, rather than negatively: e.g., that WC-groups of local and global fields are especially interesting. That's much harder to argue with. 72.152.91.213 08:49, 22 October 2007 (UTC)Plclark


 * oops, sorry about that. I've fixed the statement about the Selmer group being a subgroup of the WC group. RobHar 20:40, 22 October 2007 (UTC)

F. Châtelet
Apparently, F. Châtelet refers to François Châtelet who wrote a PhD thesis on arithmetic geometry in 1944, according to J.-L. Colliot-Thélène, Les grands thèmes de François Châtelet, Enseign. Math. 34 (1988), 387–405 --see also this). He should not be confused with François Châtelet (philosopher). --Omnipaedista (talk) 07:48, 19 February 2010 (UTC)