Elliptic fibration with a section

Assume the Axiom of Choice, inherited from the generic-fiber and group-law suppliers. An elliptic fibration with section is a flat genus-one fibration $\pi :Y\to B$ together with a specified regular section $s:B\to Y$, meaning $\pi \circ s={\mathrm{id}}_B$. A rational section is a section of the generic fiber over the function field $K(B)$, equivalently a rational map $B⇢Y$ over $B$ defined on a dense open subset; it need not extend to a regular section on all of $B$. Write $F=K(B)$, $E=Y_{\eta }$, and $O=s(\eta )\in E(F)$. The pointed generic fiber is an elliptic curve over $F$ with its canonical regular commutative group law: over every field extension $F^{\mathrm{\prime }}\mathrm{/}F$, the sum of $P,Q\in E(F^{\mathrm{\prime }})$ is the unique point $R$ for which the divisor $[R]$ is linearly equivalent to $[P]+[Q]-[O]$; the inverse of $P$ is the unique point in the class $2[O]-[P]$. The Mordell–Weil group is the abelian group $E(K(B))$, equivalently the group of rational sections with these operations performed on the generic fiber.

Flat genus-one fibration

The Axiom of Choice

A pointed genus-one curve has a regular group law over its ground field

Properness survives arbitrary base change

Support dimension under field extension

flat local ascent of regularity

Regular equals smooth over a perfect field

Flat field extension commutes with coherent cohomology

Geometric properties of fibres

T. Weigand, TASI Lectures, §§3.1 and 8.2

D. R. Morrison and D. S. Park, §3.1

Here the function field $F=K(B)$ is a finitely generated extension of $\mathbb{C}$, hence of characteristic zero; it is not assumed algebraically closed. The generic-fiber group law follows from the actual fibration data as follows. The scheme $E$ is proper and finite type over $F$ by Properness survives arbitrary base change, and its geometric generic fiber is a smooth geometrically integral genus-one curve by Flat genus-one fibration. The closure test in Geometric properties of fibres therefore gives geometric integrality of $E$, and Support dimension under field extension, applied to its structure sheaf, gives $\dim E=1$. To check smoothness over $F$ itself, take any affine chart $\mathrm{Spec}A\subset E$. The ring map $A\to A{\otimes }_F\bar{F}$ is faithfully flat. At each prime of $A$, choose a prime above it; the corresponding local map is flat local between Noetherian rings with regular target, because $E_{\bar{F}}$ is smooth. The descent clause of flat local ascent of regularity makes its source regular. Thus $E$ is regular, and Regular equals smooth over a perfect field makes it smooth over the perfect field $F$. Finally Flat field extension commutes with coherent cohomology identifies $H^1(E,{\mathcal{O}}_E){\otimes }_F\bar{F}$ with the one-dimensional $H^1(E_{\bar{F}},\mathcal{O})$, so $E$ has genus one over $F$. The section gives the $F$-rational point $O$. All hypotheses of A pointed genus-one curve has a regular group law over its ground field now hold over $F$, and that lemma constructs multiplication and inversion as regular $F$-morphisms with exactly the displayed divisor-class rules. This establishes the full group-valued clause without assuming a group scheme over the base.

For the equivalence with rational maps, restricting a section on a dense open to $\eta$ gives an $F$-point. Conversely, an $F$-point of the finite-type $B$-scheme $Y$ is described in affine neighbourhoods by finitely many coordinate functions in $K(B)$ satisfying finitely many equations. Clearing their finitely many denominators on an affine neighbourhood of $\eta$ gives a morphism over $B$ on a dense open subset. The base-coordinate maps show that its composite with $\pi$ is the identity there. Two extensions of the same generic point agree after shrinking to a common dense open, since the target is separated and the source is integral. Thus rational sections modulo shrinking are exactly $E(F)$, and the group law transfers through this bijection. No extension across every point of $B$, group law on singular fibers, or finite-generation theorem for $E(F)$ is asserted here. The Axiom of Choice is The Axiom of Choice; the group-law lemma explicitly supplies dependent choice from it for the coherent-cohomology argument.