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Is there an analog for the Tate-Shafarevich group for hyperelliptic curves?

References to such an analog would be nice if one exists.

EDIT: Referring to Noam Elkies' comment, are there any finiteness conjectures for such an analog?

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There are Tate-Shafarevich groups for abelian varieties of any dimension. If $C$ is a curve of genus 2 or more (whether or not it is hyperelliptic), strictly speaking $C$ doesn't have a Tate-Shafarevich group, but its Jacobian $J(C)$ does, and one sometimes calls that group the "Tate-Shafarevich group of the curve $C$" by abuse of terminology. –  Noam D. Elkies May 5 '12 at 3:25
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Dear Eugene, Yes, it is conjectured that Sha of any abelian variety over a number field is finite. Regards, Matthew –  Emerton May 5 '12 at 6:36
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1 Answer

up vote 5 down vote accepted

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.

This definition and some analysis of the set can be found in the very interesting paper B. Mazur: On the passage from local to global in numer theory, III §15.

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.

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Actually, in a lot of cases one can still get an abelian group structure on these pointed sets. See the work of Kneser and Borovoi. For example, in "Abelian Galois Cohomology of Reductive Groups", Borovoi shows how to use the functor $H^1_{ab}$ to give an abelian group structure on $Ш(G)$ for reductive groups $G$ over fields of characteristic 0. –  Dror Speiser May 5 '12 at 9:46
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