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For an integral Minimal Weierstrass equation, let $E_0({\mathbb{Q}}_p)$ be the subgroup of points whose reduction is a smooth point of the reduced cubic. Its reduction is a homomorphism to the smooth-locus group of a singular Weierstrass cubic, or to the whole reduced elliptic curve in the good-reduction case. Define the formal kernel of a minimal Weierstrass equation byIt consists of the identity and the points reducing to the identity at infinity. For a nonidentity affine point this is equivalent to $v_p(x)=-2r$ and $v_p(y)=-3r$ for some positive integer $r$.

The local parameter $t=-x/y$, with $t(O)=0$, gives the group isomorphismwhere $\hat{E}$ is the formal group of an elliptic curve. To see the local coordinates, put $w=-1/y$ in the Weierstrass equation of an elliptic curve. The resulting relation is $w=t^3+a_{1}tw+a_2{t}^{2}w+a_3{w}^2+a_{4}t{w}^2+a_6{w}^3$, and its unit linear coefficient gives a unique integral series $w(t)=t^3+O(t^4)$. Thus $x=t/w$ and $y=-1/w$ have valuations $-2v_p(t)$ and $-3v_p(t)$. The chord-and-tangent group law expressed in this parameter is precisely the integral formal group law $\hat{E}$. This description includes bad reduction; only the smooth identity neighborhood is involved.

Let $C$ be the projective cubic of a Weierstrass equation of an elliptic curve, allowing its elliptic-curve discriminant to vanish, and let $O=[0:1:0]$. Use only the smooth locus of a variety $C_{\mathrm{sm}}(K)$: the singular point, if present, is excluded. For $P,Q$, intersect their chord with $C$, using the tangent if $P=Q$ and counting intersection multiplicity. If the third intersection is $R$, define $P+Q=-R$, wherein general Weierstrass equation of an elliptic curve coordinates. The line through $R$ and $O$ gives this reflection. A vertical chord gives $P+(-P)=O$, and the tangent at $O$ meets $C$ three times at $O$, so $O$ is the identity. The construction is symmetric in $P,Q$. It stays in the nonsingular locus: a line through a singular point has intersection multiplicity at least two there, and therefore cannot also contain two smooth intersections counted with multiplicity.

For the nonsingular case, prove associativity by transporting a known abelian group law. The Abel-Jacobi map of a genus-one curveis bijective over an algebraic closure. Indeed, the Riemann-Roch theorem in genus one says that every divisor class of degree one has a unique effective representative consisting of one point: existence follows from $\ell (D)=1$, and uniqueness follows because two distinct representatives would produce a degree-one map to the projective line, impossible for a genus one curve. Subtracting $O$ gives the claimed bijection.

Any line section represents the same divisor class as $3O$, including tangencies. Thus $P+Q+R\sim 3O$ as divisors on an algebraic curve, while the vertical line gives $R+(-R)+O\sim 3O$. HenceAddition in the Picard group is associative, soThe chord-and-tangent group law is defined over $K$, so the group operation restricts to the $K$-rational points.

For completeness, a singular Weierstrass equation of an elliptic curve produces the smooth-locus group of a singular Weierstrass cubic, not an elliptic curve. Over an algebraic closure, its normalization of an algebraic curve is the projective line; deleting the two preimages of a nodal crossing gives the multiplicative algebraic group, while deleting the single preimage of a cusp gives the additive group. For example, on $y^2=x^3$, the coordinate $u=x/y$, with $u(O)=0$, makes the smooth-locus law addition. On $y^2=x^2(x+1)$ in characteristic different from two, put $t=y/x$ and $z=(t+1)/(t-1)$, with $z(O)=1$; the chord relation gives $z(P)z(Q)z(R)=1$, so the group law is multiplication of $z$. A nonsplit node gives the corresponding form of the multiplicative algebraic group over $K$. These descriptions also establish the singular-case group laws.