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For an elliptic curve over a local field with integral Minimal Weierstrass equation, let $E_0(K)$ consist of points with nonsingular reduction and let $E_1(K)$ be the kernel of $E_0(K)\to {\tilde{E}}_{\mathrm{ns}}(k)$. This definition also applies at a prime of bad reduction of an elliptic curve. The point at infinity is always smooth. With $t=-x/y$ and $w=-1/y$, the Weierstrass equation of an elliptic curve becomesThe unit linear coefficient in $w$ gives a unique integral formal power series $w(t)=t^3+O(t^4)$. Then $x=t/w$ and $y=-1/w$, so the nonidentity points of $E_1(K)$ have $v(x)=-2r$ and $v(y)=-3r$ for $r=v(t)>0$. The local parameter $t$ bijects $E_1(K)$ with ${\mathfrak{m}}_K$ and turns its addition into the formal group of an elliptic curve. At good reduction this agrees with the kernel of reduction of an elliptic curve.

Choose an integral Minimal Weierstrass equation over ${\mathbb{Z}}_p$. Let $E_0({\mathbb{Q}}_p)$ be the points with nonsingular reduction, and let $E_1({\mathbb{Q}}_p)$ be those reducing to $O$. This distinction matters at bad reduction of an elliptic curve: the whole singular reduced cubic is not a group. The smooth locus is a group under the induced elliptic curve group law, and Hensel's lemma lifts each of its rational points, givingThe formal kernel of a minimal Weierstrass equation identifies $E_1({\mathbb{Q}}_p)$ with $p{\mathbb{Z}}_p$ equipped with $F$. Indeed, the integral series $w(t)$ converges there, and $x=t/w(t)$, $y=-1/w(t)$ give the unique point for each nonzero parameter. If $v_p(t)=r>0$, then $v_p(x)=-2r$ and $v_p(y)=-3r$; the parameter zero gives $O$. Conversely a point reducing to $O$ has these local coordinates of positive valuation, so is recovered by the series.

Define $E_m$ for $m\ge 1$ by $t\in p^m{\mathbb{Z}}_p$. The linear part of the formal group law is addition, and the higher terms have degree at least two. Hence the filtration of elliptic-curve points over a local field hasAt good reduction, $E_0=E$ and $E/E_1\cong \tilde{E}({\mathbb{F}}_p)$. At bad reduction, there is also the finite quotient $E/E_0$: $E({\mathbb{Q}}_p)$ is a compact group because the curve is projective, and $E_0$ is an open subgroup, so its index of a subgroup is finite. Together these quotients and the formal filtration describe the group, including the extra components at bad primes.

The formal logarithm makes the deepest part explicit. Write $L(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$. Since its derivative is the integral invariant differential of a formal group law, $n{c}_n\in {\mathbb{Z}}_p$. ConsequentlyThe inequality $v_p(n)\le (n-1)/(p-1)$ shows that for an integer $m>1/(p-1)$ every nonlinear term on $p^m{\mathbb{Z}}_p$ has strictly higher valuation than the linear term. After scaling, $L$ is the identity plus a strictly contracting series; solving $L(t)=u$ by successive correction gives its inverse formal group exponential. Thus the deep logarithm subgroup of a formal group givesOne may take $m=1$ for odd $p$ and $m=2$ for $p=2$. In particular, $E({\mathbb{Q}}_p)$ contains an open subgroup isomorphic to the additive p-adic integers, and the remaining information is in finite quotients and their group extensions.

An elliptic curve over $\mathbb{Q}$ has good reduction at $p$ if it admits an integral Weierstrass equation of an elliptic curve over ${\mathbb{Z}}_p$ whose reduced projective cubic is a smooth algebraic curve over ${\mathbb{F}}_p$. Equivalently, its Minimal Weierstrass equation has a unit elliptic-curve discriminant, or its smooth proper model is an elliptic curve over the valuation ring. The existence of an appropriate integral model matters: a nonminimal equation can have discriminant divisible by $p$ even when the curve has good reduction. Equivalently,

Choose an integral Minimal Weierstrass equation whose discriminant is a $p$-adic unit. Given $P=[X:Y:Z]\in E({\mathbb{Q}}_p)$, scale the coordinates so that $X,Y,Z\in {\mathbb{Z}}_p$ and at least one is a unit. DefineAnother primitive representative differs by a unit scalar, so the projective reduction is well defined. The reduced cubic is smooth by good reduction.

To prove the group homomorphism property, view this smooth projective cubic with its section $O$ as the integral elliptic model $\mathcal{E}/{\mathbb{Z}}_p$. Its addition law is defined over ${\mathbb{Z}}_p$, by adding degree-zero divisor classes, and thus is a regular morphism $\mathcal{E}\times \mathcal{E}\to \mathcal{E}$. Properness identifies $E({\mathbb{Q}}_p)$ with the integral sections just described. Reducing the addition morphism is the group operation of the smooth special fiber. ConsequentlyThis is the good-reduction map respects elliptic addition argument; it covers tangent, vertical and infinity cases where reducing a single affine slope formula would involve division by a nonunit. Smoothness also makes reduction surjective: at each reduced point a partial derivative is a unit, and the Hensel lemma lifts that point in a local affine chart.

Bad reduction of an elliptic curve must be tested on a Minimal Weierstrass equation, not inferred merely from primes dividing the displayed elliptic-curve discriminant. For an odd prime $p$, write $v_p(D)=2k+ϵ$ with $ϵ\in \{0,1\}$. The coordinate change $x=p^{2k}X$, $y=p^{3k}Y$ gives the same form with parameter $D/p^{2k}$. If $ϵ=0$, that parameter is a unit at $p$ and its discriminant is a unit, so reduction is good.

If $ϵ=1$, the new discriminant valuation is six. Discriminants of isomorphic Weierstrass models differ by a twelfth power under a coordinate change, so their valuations differ by a multiple of twelve. An integral model has nonnegative discriminant valuation; consequently a model of valuation six cannot be replaced by one of valuation zero. Reduction is bad. At $p=2$, oddness of $D$ makes the original discriminant valuation six, and the same argument gives bad reduction of an elliptic curve there as well. ThusThese are the bad primes of a congruent number curve with odd parameter. If $D$ is squarefree this simplifies to the primes dividing $2D$. Squarefreeness is not stated in the PDF: for example, $D=9$ gives a curve isomorphic to $E_1$ by $x=9X,y=27Y$, so it has good reduction at three although the displayed discriminant is divisible by three.

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.

An elliptic curve has good reduction over $K$ if it admits an integral Weierstrass equation of an elliptic curve with unit elliptic-curve discriminant. Equivalently, the reduction of its Minimal Weierstrass equation is a nonsingular cubic over the residue field $k$. Its smooth integral projective model is a group scheme with the point $O$ as identity.

Choose homogeneous coordinates in the valuation ring ${\mathcal{O}}_K$ with at least one coordinate a unit, and reduce them modulo its maximal ideal ${\mathfrak{m}}_K$. This defines $\mathrm{red}:E(K)\to \tilde{E}(k)$. The chord-and-tangent group law extends over the smooth integral model, so reduction is a group homomorphism. For any point of $\tilde{E}(k)$, choose an affine projective chart containing it. Smoothness means some partial derivative of its equation is nonzero in $k$. Lift the other coordinate arbitrarily and apply the Hensel lemma to the coordinate with unit derivative. This constructs a point of $E(K)$ above the prescribed point, proving the surjectivity of good reduction over a local field.

The kernel of reduction of an elliptic curve consists precisely of points reducing to $O$. The parameter $t=-x/y$ at $O$ gives a bijection between this kernel and ${\mathfrak{m}}_K$: the integral formal-coordinate series solve the equation uniquely near $O$. Addition in these coordinates is the formal group of an elliptic curve, not ordinary addition in ${\mathfrak{m}}_K$. ThusHere the group operation on the last expression is its formal group law.

The curve has good reduction when it admits a Minimal Weierstrass equation over ${\mathcal{O}}_K$ whose discriminant is a unit, so reduction modulo $\pi$ is a nonsingular elliptic curve $\tilde{E}/k$. Write a point of $E(K)$ in primitive projective coordinates $(X:Y:Z)$ over ${\mathcal{O}}_K$. Reducing all three coordinates definesPrimitivity ensures that the reduced triple is not zero. The addition morphism on the smooth Weierstrass model reduces to the addition morphism on $\tilde{E}$; equivalently, this follows from the valuative criterion for properness for the smooth proper group scheme. Therefore $\mathrm{red}(P+Q)=\mathrm{red}(P)+\mathrm{red}(Q)$.

A Minimal Weierstrass equation for $E/{\mathbb{Q}}_p$ is a Weierstrass equation of an elliptic curve with coefficients in ${\mathbb{Z}}_p$ whose discriminant has minimum $p$-adic valuation among all integral equations for $E$ related by admissible changes of variables.

Let $P\in E({\mathbb{Q}}_p)$ and choose projective coordinates $[X:Y:Z]$ with $X,Y,Z\in {\mathbb{Z}}_p$ and at least one coordinate a unit. Reducing the coordinates modulo $p$ givesMultiplying the primitive coordinates by a unit does not alter this point, so this defines the reduction map $E({\mathbb{Q}}_p)\to \bar{E}({\mathbb{F}}_p)$.