The Weil–Châtelet group is an object in algebraic geometry and arithmetic geometry that classifies principal homogeneous spaces (also called torsors) of an abelian variety over a given field.
Definition
Let $K$ be a field with separable closure $\overline{K}$ and absolute Galois group $G_K=\mathrm{Gal}(\overline{K}/K)$. For an abelian variety $A$ defined over $K$, the Weil–Châtelet group of $A$ over $K$ is
$$
\mathrm{WC}(A/K)=H^{1}(G_K, A(\overline{K})),
$$
the first Galois cohomology group with coefficients in the group of $\overline{K}$-rational points of $A$. Elements of $\mathrm{WC}(A/K)$ correspond bijectively to isomorphism classes of principal homogeneous spaces (torsors) under $A$ that become isomorphic to $A$ over $\overline{K}$.
Historical background
The concept originates from André Weil’s work on abelian varieties and the arithmetic of algebraic curves (1930s–1940s). Claude Châtelet later studied the same cohomological classification for elliptic curves and more general abelian varieties, leading to the eponymous designation “Weil–Châtelet group.”
Key properties
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Exact sequence – For any short exact sequence of $G_K$-modules $0\to A\to B\to C\to 0$ there is an associated long exact sequence of Galois cohomology; in particular, the Weil–Châtelet group fits into the exact sequence
$$ 0 \to A(K) \to B(K) \to C(K) \xrightarrow{\delta} \mathrm{WC}(A/K) \to \mathrm{WC}(B/K) \to \mathrm{WC}(C/K) . $$ -
Relation to the Tate–Shafarevich group – The Tate–Shafarevich group $\Sha(A/K)$ is defined as the subgroup of $\mathrm{WC}(A/K)$ consisting of torsors that have points everywhere locally (i.e., over all completions of $K$). Thus $\Sha(A/K) \subseteq \mathrm{WC}(A/K)$.
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Finiteness conjectures – Over a number field $K$, it is conjectured (as part of the Birch and Swinnerton-Dyer conjecture) that $\Sha(A/K)$ is finite; however, the full Weil–Châtelet group is typically infinite.
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Behaviour under field extensions – For a finite extension $L/K$, there are restriction and corestriction maps
$$ \mathrm{Res}{L/K}: \mathrm{WC}(A/K) \to \mathrm{WC}(A_L/L),\qquad \mathrm{Cor}{L/K}: \mathrm{WC}(A_L/L) \to \mathrm{WC}(A/K), $$
satisfying the usual functoriality properties. -
Computational aspects – In practice, explicit computation of $\mathrm{WC}(A/K)$ is difficult. For elliptic curves over local fields, the group is known to be isomorphic to the quotient $K^{\times}/N_{L/K}(L^{\times})$ for certain extension fields $L$ (via the Weil–Châtelet isomorphism for the multiplicative group). Over global fields, one often works with its Selmer subgroups, which are finite and effectively computable.
Examples
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Elliptic curves – If $E/K$ is an elliptic curve, $\mathrm{WC}(E/K)$ classifies genus‑1 curves over $K$ that acquire a rational point over $\overline{K}$ and are equipped with a simply transitive action of $E$. The “principal homogeneous spaces” are precisely the curves of genus 1 without a $K$-rational point.
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Trivial cases – If $A$ has a $K$-rational point (i.e., $A(K) eq\emptyset$), the trivial torsor corresponds to the identity element of $\mathrm{WC}(A/K)$. For the additive group $\mathbb{G}_a$ over a perfect field, the first cohomology group vanishes, so $\mathrm{WC}(\mathbb{G}_a/K)=0$.
Applications
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Descent theory – The Weil–Châtelet group underlies the method of descent on abelian varieties, providing the cohomological framework for constructing covering spaces that help bound the Mordell–Weil rank.
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Obstructions to the Hasse principle – Non‑trivial elements of $\Sha(A/K)\subseteq \mathrm{WC}(A/K)$ give explicit counter‑examples to the Hasse principle for genus‑1 curves.
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Birch and Swinnerton‑Dyer conjecture – The order of $\Sha(A/K)$ appears in the conjectural formula for the leading term of the $L$-function of $A$ at $s=1$.
References
- A. Weil, Variétés abéliennes et courbes algébriques, Hermann, 1948.
- J.-P. Serre, Galois Cohomology, Springer, 1997.
- J. Tate, “The Arithmetic of Elliptic Curves,” Invent. Math. 23 (1974), 179–206.
- J. W. S. Cassels, Local Fields, Cambridge University Press, 1986 (chapter on cohomology of abelian varieties).
