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{title2=$\mathrm{WC}(A/K)=H^1(K,A)$}
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For an <abelian variety> $A$ over a field $K$, its Weil–Châtelet group is the <Galois cohomology> group $H^1(K,A(K^{\rm sep}))$, classifying torsors under $A$. Such an <algebraic-group torsor> becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an <elliptic curve>, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the <Tate–Shafarevich group>.


 Weil–Châtelet group (source code)