Formal group law

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfying the identity, commutativity, and associativity axioms. A homomorphism from $F$ to $G$ is a series $\theta (T)\in TR[[T]]$ satisfying $\theta (F(X,Y))=G(\theta (X),\theta (Y))$.

For a one-dimensional formal group law, write $f(T)={\partial }_{Y}F(T,0)$. Its invariant differential is $dT/f(T)$. Differentiating associativity gives ${\partial }_{Y}F(X,Y)f(Y)=f(F(X,Y))$, which expresses invariance under translation. In characteristic zero, integrating this differential constructs the formal logarithm.

An isomorphism from formal group law $F$ to $G$ over $R$ is a formal series $h(T)=a_{1}T+a_2{T}^2+\cdots$ with $a_1\in R^{\ast }$ and $h(F(X,Y))=G(h(X),h(Y))$. Coefficient recursion gives a compositional inverse over $R$. An isomorphism of formal series need not converge on an arbitrary prescribed analytic domain; convergence must be checked separately.

Every one-dimensional formal group law has a unique formal inverse $i_F(X)=-X+O(X^2)$ satisfying $F(X,i_F(X))=0$. Comparing coefficients recursively determines it degree by degree.

The formal additive group has law $F_a(X,Y)=X+Y$. Its multiplication-by-$n$ series is $[n{]}_a(X)=nX$.

The formal multiplicative group has law $F_m(X,Y)=X+Y+XY$, obtained by writing a multiplicative coordinate as $1+X$. Its multiplication-by-$n$ series is

Over an algebra over a field whose scalar field is $\mathbb{Q}$, the series $\exp (X)-1$ defines an isomorphism from the formal additive group to the formal multiplicative group, with inverse $\log (1+X)$.

A morphism $h(T)=uT+O(T^2)$ of one-dimensional formal group laws over a ring is an isomorphism when $u$ is a unit. Recursive coefficient comparison constructs its compositional inverse, and the morphism identity shows that this inverse is also a formal-group morphism.

For a one-dimensional formal group law $F$ over a ring $R$, the multiplication series satisfies $[n{]}_F(T)=nT+O(T^2)$. It is an isomorphism exactly when $n$ is a unit of $R$.

Let $R$ be a complete discrete valuation ring of residue characteristic of a field $p$, with maximal ideal $\mathfrak{m}$, and let $F$ be a one-dimensional formal group law over $R$. Its multiplication series is $[n{]}_F(T)=nT+O(T^2)$. If $p\nmid n$, the invertible morphism criterion for formal group laws constructs its inverse with coefficients in $R$. Both formal power series converge on $\mathfrak{m}$, since their coefficients are integral and the powers of an element of $\mathfrak{m}$ tend to zero. Their compositions are the identity there. Thus multiplication by $n$ is bijective, including when $p=2$ and $n$ is odd. Every finite-order element of $F(\mathfrak{m})$ consequently has order of a group element equal to a prime power with prime $p$.

Over a characteristic-zero field, a one-dimensional commutative formal group law has a unique strict isomorphism to the additive formal group, called its formal logarithm. It satisfies ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$.

For an integral one-dimensional formal group law over a finite extension of the p-adic field, the inverse of ${\partial }_{Y}F(T,0)$ has integral coefficients because its constant term is one. Integrating this invariant differential gives ${\log }_F(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$ with $n{c}_n$ integral. For $v_K(\pi )=1$ and $e=v_K(p)$ the displayed bound follows. The estimate $v_p(n)\le (n-1)/(p-1)$ makes the nonlinear part strictly contracting on ${\pi }^s{\mathcal{O}}_K$ for $s>e/(p-1)$, yielding the deep logarithm subgroup of a formal group.

Let $F$ be a formal group law over the valuation ring of a finite extension $K/{\mathbb{Q}}_p$, with $v_K(\pi )=1$ and $e=v_K(p)$. If $m>e/(p-1)$, its formal logarithm is an isomorphism $F({\pi }^m{\mathcal{O}}_K)\to ({\pi }^m{\mathcal{O}}_K,+)$. Indeed, its scaled series is $U+\pi R(U)$ with $R$ integral and convergent on ${\mathcal{O}}_K$; solving by contraction produces the formal group exponential. The subgroup has index $q^{m-1}$ in $F(\pi {\mathcal{O}}_K)$, where $q$ is the residue-field order.

Let $E/{\mathbb{Q}}_p$ have good reduction, and choose an integer $m>1/(p-1)$. Its filtration of elliptic-curve points over a local field has a torsion-free group $E_m$ identified by the formal logarithm with $p^m{\mathbb{Z}}_p$. The quotient group $G=E({\mathbb{Q}}_p)/E_m$ is finite. For $g\in G$ of order of a group element $N$, choose a lift $P$. There is a torsion point of an elliptic curve in this coset exactly when the displayed condition holds. Indeed, it is equivalent to the existence of $Q\in E_m$ with $[N]Q=[N]P$. Then $P-Q$ is killed by $N$; conversely every torsion representative has order $N$, since $E_m$ is torsion-free. The logarithmic division and the formal group exponential give the unique such $Q$. Changing the lift changes the logarithm by an element of $N{p}^m{\mathbb{Z}}_p$, so the criterion is well-defined. This finite-coset test includes residue-characteristic torsion; only the prime-to-p torsion lifts uniquely at good reduction are automatic.

For a one-dimensional commutative integral formal group law, the formal logarithm identifies $F(4{\mathbb{Z}}_2)$ with $4{\mathbb{Z}}_2$. This is an index-two subgroup of $F(2{\mathbb{Z}}_2)$. Choose a topological generator $h$ of the subgroup and an element $v$ outside it, and write $2v=ch$. If $c$ is odd, $v$ generates the whole group. If $c$ is even, $v-(c/2)h$ has order two and splits off a cyclic factor. The formal additive group and formal multiplicative group realize the two cases.

For odd $p$, a one-dimensional formal group law over ${\mathbb{Z}}_p$ has a formal logarithm ${\log }_F(T)=T+{\sum }_{j\ge 2}{c}_j{T}^j/j$ with $c_j\in {\mathbb{Z}}_p$. This follows by integrating its integral invariant differential. For $t\in p{\mathbb{Z}}_p\setminus \{0\}$, $v_p(c_j{t}^j/j)>v_p(t)$ for $j\ge 2$. Hence the series converges and $v_p({\log }_F(t))=v_p(t)$, so its kernel is zero. As it is a homomorphism into a characteristic-zero additive group, there is no nonzero torsion. In particular, at an odd prime of good reduction over $\mathbb{Q}$, reduction injects the whole rational torsion subgroup, including its $p$-primary part.

The formal group exponential is the compositional inverse of the formal logarithm. On a sufficiently small neighborhood over a characteristic-zero local field, both series converge and identify the formal group with the additive group.

The local parameter $z=-x/y$ at the identity of a Weierstrass curve expresses addition as a formal group law. Over a local field it describes points reducing to the identity.

For a Weierstrass equation of an elliptic curve, use the local parameter $t=-x/y$ and put $w=-1/y$. The formal kernel of a minimal Weierstrass equation construction gives $w=t^3+a_1{t}^4+(a_1^2+a_2)t^5+(a_1^3+2a_1{a}_2+a_3)t^6+O(t^7)$. Substitution into the elliptic curve group law yields the displayed initial coefficients and the formal inverse $-t/(1-a_{1}t-a_{3}w(t))$. In characteristic zero, the invariant differential on an elliptic curve becomes $(1+a_{1}t+(a_1^2+a_2)t^2+O(t^3))dt$, giving the formal logarithm $t+a_1{t}^2/2+(a_1^2+a_2)t^3/3+O(t^4)$. The addition coefficients are integral polynomial expressions and hence also apply in positive characteristic of a field.

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.

A nonidentity rational point in the formal kernel of a minimal Weierstrass equation at two distinct primes has infinite order of a group element. Indeed, the prime-to-residue-characteristic multiplication on a formal group implies that a finite order in the kernel at $p$ must be a power of $p$. Membership in the kernel at a different prime $q$ also forces a power of $q$. The only common possibility is order one, contradicting nonidentity. For an integral model, negative valuations of the affine coordinates provide a convenient kernel-membership test.

For an integral short equation $y^2=x^3+ax+b$, negation acts on $t=-x/y$ by $t\mapsto -t$. Thus the integral formal multiplication series is odd: $[2](t)=2t+t^{3}g(t)$. If $v_2(t)=s\ge 1$, the leading term has valuation $s+1<3s$, so successive doubling never kills a nonzero point. Odd multiplication has unit linear coefficient. Combined with the formal logarithm at odd primes, this proves torsion-freeness of the formal identity neighbourhood at every prime for a short equation. The short-model hypothesis matters at two.

For $y^2=x^3+ax+b$, put $t=-x/y$ and $w=-1/y$. Then $x=t/w$, $y=-1/w$, and the curve equation becomesThere is a unique solution $w(t)\in t^{3}K[[t]]$, and points in the kernel of reduction correspond to parameters of positive valuation.

An isogeny $\varphi :E\to E^{\prime }$ sends the identity to the identity, so substituting the formal coordinates into $t^{\prime }=-x^{\prime }/y^{\prime }$ gives a series $f(t)\in tK[[t]]$. Compatibility with the elliptic-curve group laws givesmaking $f$ a morphism of formal group laws.

For an elliptic curve over a nonarchimedean local field, $E_0$ consists of points with nonsingular reduction, $E_1$ is the kernel of reduction, and $E_r$ for $r\ge 1$ consists of points whose formal parameter has valuation at least $r$. The successive quotients for $r\ge 1$ are copies of the additive residue-field group.