Is there an analog for the TateShafarevich 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?
Is there an analog for the TateShafarevich 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? 


There are TateShafarevich 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 TateShafarevich "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 nonabelian 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 TateShafarevich 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. 

