The chord-tangent group law on a smooth short Weierstrass cubic

The cubic $C$ is smooth over $k$: on the affine chart $Z=1$ a common zero of $\mathrm{\partial }F\mathrm{/}\mathrm{\partial }x=-3x^2-a$ and $\mathrm{\partial }F\mathrm{/}\mathrm{\partial }y=2y$ would force $y=0$, $3x^2=-a$ and $b-2x^3=0$, hence $4a^3+27b^2=0$, and in the chart $Y=1$ the gradient at $O$ is nonzero because $\mathrm{\partial }(Z-X^3-aX{Z}^2-b{Z}^3)\mathrm{/}\mathrm{\partial }Z=1$ at $O$; the Jacobian criterion is compatible with field extension, so $C$ is smooth over every extension of $k$. If $C_{\bar{k}}$ were reducible, its components would be plane curves of positive degrees summing to $3$, and by [F2] they would meet in a nonempty divisor, at whose points $C$ would not be regular, a contradiction; hence $C$ is geometrically integral and, being a closed subscheme of ${\mathbf{P}}_k^2$, proper of dimension one over $k$. Its genus is one by [F2], so $C$ is a curve in the sense of [F1] of genus one.

Since $O\in C(k)$, the class ${\mathcal{O}}_C(3O)$ has degree $3$, and the assignment $\phi (P)=[{\mathcal{O}}_C(P-O)]$ defines a map $C(\bar{k})\to {\mathrm{Pic}}^0(C_{\bar{k}})$: the difference of two degree-one divisors has degree zero. It is bijective. Indeed, for $[\mathcal{L}]\in {\mathrm{Pic}}^0(C_{\bar{k}})$ the sheaf $\mathcal{L}\otimes {\mathcal{O}}_C(O)$ has degree one and $l=\mathrm{deg}=1$ by Riemann-Roch and triviality of the canonical bundle in [F3], so it admits a nonzero section whose divisor $P$ is effective of degree one; then $[\mathcal{O}(P-O)]=[\mathcal{L}]$. Uniqueness holds because $l(\mathcal{O}(P))=1$, so the effective divisor of degree one in a degree-one class is unique.

Let $P,Q\in C(\bar{k})$ and let $\mathrm{\ell }$ be the line through $P$ and $Q$, tangent at $P$ when $P=Q$; write $P+Q+R$ for the divisor of the corresponding hyperplane section, which has degree $3$ by [F2]. The restriction of $\mathcal{O}(1)$ to $C$ is isomorphic to ${\mathcal{O}}_C(3O)$: the coordinate $Z$ restricts to a section whose divisor is $3O$, since on the chart $Y=1$ the equation of $C$ is $Z=X^3+aX{Z}^2+b{Z}^3$, so the intersection with the line $Z=0$ is the point $X=Z=0$ with multiplicity three. Hence $P+Q+R\sim 3O$. For the vertical line through $R$ the intersection divisor is $R+(-R)+O$, so $R+(-R)\sim 2O$. Combining, $P+Q-(-R)\sim O$, that is, in ${\mathrm{Pic}}^0$, $\phi (-R)=\phi (P)+\phi (Q)$: the assignment of step 2.1 converts the chord-tangent operation $P+Q:=-R$ into addition in the abelian group ${\mathrm{Pic}}^0(C_{\bar{k}})$. Consequently the operation is commutative and associative, has identity $O$, and inverse $-\text{\hspace{0.17em}⁣}P=[X:-Y:Z](P)$; and in the affine chart the standard substitution of the line $y=\lambda x+\nu$ in $y^2=x^3+ax+b$ gives the displayed formulas, with $\lambda$ as in the statement when the chosen denominator is nonzero.

The operation is algebraic. On $(C\setminus \{O\})\times (C\setminus \{O\})$ with affine coordinates $(x_i,y_i)$ define $(N,D)=(y_2-y_1,x_2-x_1)$ on the first open and $(N,D)=(x_1^2+x_1{x}_2+x_2^2+a,y_1+y_2)$ on the second, and consider the morphism $(P,Q)\mapsto [D(N^2-(x_1+x_2)D^2):N((2x_1+x_2)D^2-N^2)-y_1{D}^3:D^3]$. On $D\mathrm{\ne }0$ this is the sum computed in step 3.1 with $\lambda =N\mathrm{/}D$, and on $D=0$, $N\mathrm{\ne }0$ its value is $O$, which is the sum of the inverse pair $P,Q$. The two opens cover the affine square: if the first pair $(N,D)$ vanishes then $x_1=x_2$ and $y_1=y_2$, i.e. $P=Q$, and then the second pair is $(3x_1^2+a,2y_1)$, which cannot vanish at a point of the smooth curve $C$ by step 1.1. Hence the sum is a morphism on $(C\setminus \{O\}{)}^2$; the extension at pairs involving $O$ is established next.

Over $\bar{k}$ the law extends to the full product and its group identities hold: for each $R\in C(\bar{k})$ the translation $T_R$ is a rational map from the smooth curve $C_{\bar{k}}$ to the proper scheme $C_{\bar{k}}$, hence a morphism by [F4]; $T_R$ and $T_{-R}$ are mutually inverse on a dense open and hence everywhere by [F4]; for arbitrary $(P,Q)$ and $R$ avoiding $-P$ and $Q$ the expression $(P+R)+(Q-R)$ is defined and regular near $(P,Q)$, these local morphisms agree on dense opens and hence glue to a morphism $C_{\bar{k}}\times C_{\bar{k}}\to C_{\bar{k}}$ extending the law of step 4.1. Since the group identities are identities of morphisms between reduced schemes over $\bar{k}$ and hold on the dense set of $\bar{k}$-points described in step 3.1, they hold everywhere by [F4]; the inverse $[X:-Y:Z]$ is a regular involution.

Since $C$ and $C{\times }_{k}C$ are finitely presented, [F5] descends the morphism of step 5.1 to some finite extension $L\mathrm{/}k$ inside $\bar{k}$. Its restriction to $U=(C\setminus \{O\}{)}^2$ is the $k$-defined morphism of step 4.1: this equality can be checked after the faithfully flat extension $\bar{k}\mathrm{/}L$. The two pullbacks over $L{\otimes }_{k}L$ therefore agree on $U_{L{\otimes }_{k}L}$. This open is schematically dense in $(C{\times }_{k}C{)}_{L{\otimes }_{k}L}$: on affine charts restriction to the dense open is injective before base change, and a finite principal-open cover computes its sections by a finite equalizer; tensoring over the field $k$ preserves these injections and equalizers. Thus separatedness and [F4] make the pullbacks equal even when $L{\otimes }_{k}L$ is nonreduced. The finite faithfully flat extension $L\mathrm{/}k$ is an fppf cover, so [F5] descends the law to $k$. The unit $O$ and inversion are already defined over $k$, and the group identities hold after the faithfully flat extension to $\bar{k}$, hence over $k$. The smooth proper geometrically integral curve $C$ with this law is an abelian variety.