Formal groups and local points

Fix a prime p at least five and a short integral Weierstrass model with unit discriminant over Z_p. Near O use t=−x/y and w=−1/y. Substituting x=t/w and y=−1/w into the equation gives an implicit equation for w with a unique power-series solution beginning with t³. Its coefficients are integral polynomials in the Weierstrass coefficients. Consequently x begins with t to the minus two and y with minus t to the minus three.

These are formal Laurent expansions, evaluated p-adically when t lies in pZ_p. They do not describe a real limit along a sketch of the curve. If t has positive valuation m, the leading units show that x and y have valuations minus two m and minus three m. Their projective coordinates nevertheless reduce to O. The pole is in the affine chart, while the point on the proper smooth curve remains perfectly well defined.

Writing the parameter of P+Q in terms of those of P and Q produces a commutative formal group law F(T₁,T₂)=T₁+T₂ plus terms of total degree at least two. It has a formal inverse beginning with minus T and satisfies associativity as an identity of power series. The linear term reflects the tangent space of the group. The remaining coefficients encode the elliptic geometry rather than replacing it by ordinary coordinatewise addition.

For the good model over Q_p, reduction gives a surjective group map to the finite smooth fiber. Its kernel E₁ consists precisely of the points reducing to O and is parametrized by pZ_p with the evaluated formal law. Surjectivity comes from smooth lifting over the complete discrete valuation ring. The quotient is finite, but E₁ is an infinite local group. Knowing that a point reduces to O therefore does not determine the point or imply that it is the identity.

In characteristic zero, integrating the normalized invariant differential gives a formal logarithm with leading term T. For the present Q_p model with p at least five it converges on pZ_p and identifies this formal group with the additive group pZ_p. On that domain it preserves valuation. Multiplication by n is then conjugate to ordinary multiplication by n, so its effect on valuation can be computed without evaluating many terms of the formal addition series.

The restricted base field and convergence domain matter. For ramified extensions, very small primes, or a larger neighborhood, one must establish the appropriate convergence and injectivity bounds before applying a logarithm as an isomorphism. In particular, finite-field p-torsion and formal local torsion are different questions. Prime-to-p injectivity of reduction follows directly from the invertible linear coefficient; broader claims require the local hypotheses that make the logarithm valid.