An elliptic curve over a scheme \(S\) is a proper smooth morphism \(p \colon E \to S\) with a section \(e \colon S \to E\) whose geometric fibres are connected curves of genus \(1\).
Locally on \(S\), an elliptic curve is a Weierstrass cubic: every point of \(S\) has an affine neighbourhood \(\Spec A\) over which \(E\) is the closed subscheme of \(\PP^2_A\) cut out by \[ y^2 z + a_1 xyz + a_3 y z^2 = x^3 + a_2 x^2 z + a_4 x z^2 + a_6 z^3 \] with discriminant \(\Delta \in A^\times\), and \(e\) is the point \([0:1:0]\). The fibres are elliptic curves over the residue fields, and the group laws on the fibres come from a morphism \(E \times_S E \to E\) making \(E\) a commutative group scheme over \(S\).
Over \(S = \Spec \ZZ[1/6][a, b][\Delta^{-1}]\) with \(\Delta = -16(4a^3 + 27b^2)\), the curve \(y^2 z = x^3 + a x z^2 + b z^3\) is an elliptic curve; its fibre over a point of \(S\) is the Weierstrass curve with the images of \(a\) and \(b\). Over \(\Spec \ZZ\), the equation \(y^2 z = x^3 - x z^2\) has discriminant \(64\), so it defines an elliptic curve over \(\Spec \ZZ[1/2]\) but not over \(\Spec \ZZ\).
Dependencies
D-MORSMSmooth morphisms and relative dimensionuses