Question

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?

Answer 1

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.