Reduction of an elliptic curve

Reducing a minimal integral Weierstrass equation modulo the maximal ideal gives a projective cubic over the residue field. Reducing primitive projective coordinates defines the reduction map on rational points.

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.

An elliptic curve has bad reduction at a finite place when its reduced minimal Weierstrass equation is singular, equivalently when its minimal discriminant has positive valuation.

Discriminants of isomorphic Weierstrass equations of an elliptic curve differ by a twelfth power of the scale parameter, so their valuations differ by a multiple of twelve. An integral model with good reduction has discriminant valuation zero. Thus a displayed integral model with discriminant valuation not divisible by twelve proves intrinsic bad reduction. More generally, a displayed discriminant valuation between zero and eleven is already minimal, since any smaller integral-model valuation would be nonnegative and differ by twelve. Positive valuations divisible by twelve alone do not establish bad reduction; rescaling may remove them.

For odd nonzero $D$, the intrinsic bad primes of $y^2=x^3-D^{2}x$ are exactly the displayed set. If $v_p(D)=2k$, the change $x=p^{2k}X$, $y=p^{3k}Y$ removes that even valuation from the parameter and gives unit discriminant at odd $p$. If the valuation is odd, after the same removal the discriminant valuation is six. Discriminant valuations of isomorphic Weierstrass models differ by multiples of twelve, so no integral model has unit discriminant. At two the original discriminant valuation is also six. Thus primes dividing a nonminimal displayed discriminant need not be intrinsic bad primes: $E_9$ is isomorphic to $E_1$ and has good reduction at three.

At a prime of good reduction with residue characteristic $p$, reduction is injective on the prime-to-$p$ torsion of an elliptic curve. Comparing reductions at two good primes therefore bounds rational torsion by finite point counts.

For an elliptic curve over $\mathbb{Q}$, reduction at a good prime $p$ is injective on the $\ell$-primary torsion subgroup when $\ell \ne p$: its kernel is a formal group, where multiplication by $\ell$ is bijective. Hence a rational point of order $\ell$ forces $\ell \mid \#\tilde{E}({\mathbb{F}}_p)$. If for every possible prime $\ell$ some good $p\ne \ell$ has a point count not divisible by $\ell$, rational torsion is trivial. In particular two distinct good primes $p,r$ with coprime counts, $p\nmid \#\tilde{E}({\mathbb{F}}_r)$ and $r\nmid \#\tilde{E}({\mathbb{F}}_p)$ suffice: the coprime counts exclude other primes, and the two cross conditions exclude $p,r$. This argument does not incorrectly apply prime-to-$p$ injectivity to $p$-power torsion.

Let an elliptic curve over $\mathbb{Q}$ have good reduction at distinct primes $p,r$, with point counts $N_p,N_r$. For a torsion point, its $\ell$-primary part injects into every reduction whose residue characteristic differs from $\ell$. Consequently its $\ell$-power order has exponent at most $\min (v_{\ell }(N_p),v_{\ell }(N_r))$ when $\ell \ne p,r$, at most $v_p(N_r)$ when $\ell =p$, and at most $v_r(N_p)$ when $\ell =r$. This precise form avoids assuming injectivity on residue-characteristic torsion. Counts five at two and seven at three force trivial rational torsion.

For good reduction of an elliptic curve over a complete local field of residue characteristic $p$, smoothness and the Hensel lemma make reduction surjective. Its kernel is a formal group on the maximal ideal, where multiplication by $q^m$, $q\ne p$, is bijective by prime-to-residue-characteristic multiplication on a formal group. Thus there is no nonzero $q$-power torsion in the kernel. Given a lift $P$ of a reduced point killed by $q^m$, choose the unique kernel point $R$ with $q^{m}R=q^{m}P$. Then $P-R$ is a torsion lift. This proves both surjectivity and uniqueness of torsion lifting. For finite residue field these groups are finite.

For an elliptic curve with good reduction over ${\mathbb{Q}}_2$, let $E_1$ be the reduction kernel and $E_2$ its formal subgroup with parameter in $4{\mathbb{Z}}_2$. The deep logarithm subgroup of a formal group makes $E_2$ torsion-free, while $[E_1:E_2]=2$. Thus the torsion of the reduction kernel has order at most $2$. Combined with the Hasse theorem for elliptic curves, rational torsion of order $16$ forces bad reduction at $2$.