Torsors under a specified relative group scheme on the small étale site

Let $B$ be an integral smooth complex scheme. Choose a dense open $B^{\circ }\subseteq B$ and a smooth commutative group scheme $J^{\circ }\to B^{\circ }$ of relative dimension one with a specified identity section; assume this group scheme is given, rather than inferring its existence from an arbitrary genus-one fibration. For every étale $B^{\circ }$-scheme $U\to B^{\circ }$, define $J(U):={\mathrm{Hom}}_U(U,J^{\circ }{\times }_{B^{\circ }}U),$ with restrictions by pullback and group law induced by multiplication on $J^{\circ }$. This is a sheaf of abelian groups on the small étale site. A $J$-torsor is a sheaf with a right $J$-action that is étale locally equivariantly isomorphic to $J$. The group ${\mathrm{Sha}}_{\mathrm{e}\mathrm{t}}(J^{\circ }\mathrm{/}{B}^{\circ })$ is the abelian group of isomorphism classes under contracted product.

A flat genus-one fibration $\pi :Y\to B$ restricts to a $J^{\circ }$-torsor on $B^{\circ }$ only when $Y{\mathrm{\mid }}_{B^{\circ }}$ is equipped with a $J^{\circ }$-action that is étale locally simply transitive, equivalently when it is étale locally isomorphic to $J^{\circ }$ as a $J^{\circ }$-torsor. An isomorphism of abstract genus-one curves alone does not specify this action. Representability of the resulting torsor sheaf by a scheme and extension across $B\setminus B^{\circ }$ are separate conditions.

Flat genus-one fibration

Elliptic fibration with a section

Étale morphism of schemes

The small étale site of a scheme

Jointly surjective étale families form a Grothendieck pretopology

Presheaves and sheaves of sets on a site

Representable presheaves are sheaves on the small étale site

Étale torsors and contracted products

Étale torsor classes form an abelian group

V. Braun and D. R. Morrison, §§1–2 and 4.3

T. Weigand, §8.2

Milne, Lectures on Étale Cohomology, Chapter I, §§5–6 and §11

This local definition deliberately calls the object a torsor class group, not derived $H_{\acute{e}t}^1(B^{\circ },J^{\circ })$ and not an arithmetic Tate–Shafarevich group. The research source's $H^1(B,A)$ shorthand is not identified with this sheaf-theoretic group without a separate comparison theorem and matched hypotheses. The generic pointed-curve law established in Elliptic fibration with a section supplies a group over $K(B)$. The present definition still assumes the specified smooth commutative group scheme $J^{\circ }$ on $B^{\circ }$ and its action: a generic-fiber law alone does not construct that relative group scheme or a torsor under it.