Analog for Tate-Shafarevich group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:24:01Z http://mathoverflow.net/feeds/question/96041 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96041/analog-for-tate-shafarevich-group Analog for Tate-Shafarevich group Eugene 2012-05-05T03:05:15Z 2012-05-05T06:11:48Z <p>Is there an analog for the Tate-Shafarevich group for hyperelliptic curves?</p> <p>References to such an analog would be nice if one exists.</p> <p>EDIT: Referring to Noam Elkies' comment, are there any finiteness conjectures for such an analog?</p> http://mathoverflow.net/questions/96041/analog-for-tate-shafarevich-group/96047#96047 Answer by Ralph for Analog for Tate-Shafarevich group Ralph 2012-05-05T05:39:23Z 2012-05-05T06:11:48Z <p>There are Tate-Shafarevich groups for every number field $K$ and every smooth locally algebraic group scheme $G$ over $X \setminus S$ where $X$ is the spectrum of the ring of integers in $K$ and $S$ is a finite set of places containing all infinite places. In this case, the Tate-Shafarevich "groups" (actually they are only pointed sets in general) are defined as $$Ш(G) := \ker\big(H^1(K,G) \to \prod_v H^1(K_v,G)\big)$$ where $v$ runs over all places of $K$ and $H^1$ is the non-abelian cohomology. </p> <p>This definition and some analysis of the set can be found in the very interesting <a href="http://math.stanford.edu/~lekheng/flt/mazur2.pdf" rel="nofollow">paper</a> B. Mazur: On the passage from local to global in numer theory, III §15. </p> <p>Concerning finiteness conjectures: Of interest may be Corollary 1 in Mazur's paper which states that $Ш(G)$ is finite if the Tate-Shafarevich conjecture holds for abelian varieties over $K$, i.e. $Ш(A/K)$ is finite for each abelian variety defined over $K$ and a particular group of automorphism of $G$ is descent. </p>