Good reduction of an elliptic curve

An elliptic curve has good reduction at a finite place when its reduced minimal Weierstrass equation is nonsingular, equivalently when its minimal discriminant has valuation zero.

An integral Weierstrass equation of an elliptic curve with unit elliptic-curve discriminant defines a smooth proper elliptic curve over the valuation ring. Properness identifies points over the local field with integral sections. The group law, constructed by adding degree-zero divisor classes, is a morphism over the valuation ring, and base change to the residue field preserves it. Hence reducing primitive projective coordinates is a group homomorphism, including exceptional affine addition cases.

A good reduction in characteristic $p$ is supersingular when its geometric $p$-torsion is trivial, equivalently its formal group of an elliptic curve has height two. Good inert primes of a CM elliptic curve are supersingular. The ordinary bounded-measure and torsion-Selmer formulation must then be modified, for instance using signed local conditions in the cyclotomic tower.

A good reduction in characteristic $p$ is ordinary when the geometric $p$-torsion has $p$ points, equivalently its formal group of an elliptic curve has height one. For a CM elliptic curve, a good prime split in its CM field is ordinary. One of the two roots of $X^2-a_{p}X+p$ is a $p$-adic unit, enabling the bounded ordinary p-adic L-function.

If an elliptic curve has good reduction over a local field $K$, its integral projective model is smooth. Every point over the residue field lifts by the Hensel lemma, applied in a chart with a unit partial derivative. Thus reduction gives an exact sequence $0\to E_1(K)\to E(K)\to \tilde{E}(k)\to 0$. The kernel of reduction of an elliptic curve is the maximal ideal of the valuation ring, with the formal group of an elliptic curve as its group operation.

For an elliptic curve with good reduction over a local field, the kernel of reduction consists of the points reducing to the identity. The parameter $t=-x/y$ identifies it with the maximal ideal equipped with the formal group of an elliptic curve.

On an integral Weierstrass equation of an elliptic curve with good reduction at a prime $p$, a nonidentity rational point reducing to the identity cannot have both coordinates integral at $p$. Indeed, an affine integral pair reduces to an affine point, distinct from the point at infinity. This can prove nonintegrality of a large linear combination without computing its coordinates. The nonidentity condition is essential and can often be checked using the known torsion subgroup.

At a prime of good reduction of an elliptic curve, the reduction map is a group homomorphism on all local points. If $N$ annihilates the reduced finite group, every $NP$ reduces to the identity at infinity. A finite point with integral affine coordinates reduces to an affine point, so a nonidentity $NP$ cannot have integral affine coordinates. The identity has no affine coordinates and must be treated separately.