Formal group
Encyclopedia
In mathematics
, a formal group law is (roughly speaking) a formal power series
behaving as if it were the product of a Lie group
. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic group
s) and Lie algebra
s. They are used in algebraic number theory
and algebraic topology
.
R is a power series
F(x,y) with coefficients in R, such that
The simplest example is the additive formal group law F(x, y) = x + y.
The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
More generally, an n-dimensional formal group law is a collection of n power series
Fi(x1, x2, ..., xn, y1, y2, ..., yn) in 2n variables, such that
where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on.
The formal group law is called commutative if F(x,y) = F(y,x).
Prop. If R is ZZ-torsion free then and formal group law over R is commutative.
proof. The torsion freeness gives us the exponential and logarithm which allows us to write F as F(x,y) = exp(log(x) + log(y)).
There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0.
A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that
A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)= x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b) = ab. If we "change coordinates" to make 0 the identity by putting a = 1 + x, b = 1 + y, and G = 1 + F, then we find that
F(x, y) = x + y + xy.
Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x) − 1. Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.
The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group:
Over fields of characteristic 0, formal group laws are essentially the same as finite dimensional Lie algebras: more precisely, the functor from finite dimensional formal group laws to finite dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p>0.
Examples:
If R does not contain the rationals, a map f can be constructed by extension of scalars to R⊗Q, but this will send everything to zero if R has positive characteristic. Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R⊗Q, and then proving that the coefficients of the corresponding formal group over R⊗Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W(R) of Witt vectors, and reduces to R at the end.
of a group and to the universal enveloping algebra
of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups.
For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes messier.
Suppose that F is a (1-dimensional) formal group law over R. Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra
H constructed as follows.
Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.
elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.
This makes F into a functor
from commutative R-algebras S to groups.
We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups. For example, this allows us to define F(Zp) with values in the p-adic numbers.
The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and ηg = 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form
for nilpotent elements x. In particular we can identify the group-like elements of H⊗S with the nilpotent elements of S, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S).
for some non-negative integer h, called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞.
The height of a one dimensional formal group law over a field of characteristic p > 0 is defined to be the height of its multiplication by p map.
Two one dimensional formal group laws over an algebraically closed field of characteristic p > 0 are isomorphic if and only if they have the same height, and the height can be any positive integer or ∞.
Examples:
be
for indeterminates
and we define the universal
ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:
The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, ... (where ci,j has degree 2(i + j − 1)). Daniel Quillen proved that the coefficient ring of complex cobordism
is naturally isomorphic as a graded ring to Lazard's universal ring, explaining the unusual grading.
in the category of formal scheme
s.
Formal groups and formal group laws can also be defined over arbitrary schemes
, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object.
The moduli space of formal group laws is a disjoint union of infinite dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite dimensional groupoid of coordinate changes.
Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations. For example, the Serre-Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations.
A formal group is sometimes defined as a cocommutative Hopf algebra
(usually with some extra conditions added, such as being pointed or connected). This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring.
Some authors use the term formal group to mean formal group law.
More generally we can allow e to be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p. All the group laws for different choices of e satisfying these conditions are strictly isomorphic.
For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law.
There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field.
This construction was introduced by , in a successful effort to isolate the local field
part of the classical theory of complex multiplication
of elliptic function
s. It is also a major ingredient in some approaches to local class field theory (e.g., Michiel Hazewinkel,
Local class field theory is easy.
Advances in Math. 18 (1975), no. 2, 148–181).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a formal group law is (roughly speaking) a formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
behaving as if it were the product of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s) and Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s. They are used in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
.
Definitions
A one-dimensional formal group law over a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R is a power series
F(x,y) with coefficients in R, such that
- F(x,y) = x + y + terms of higher degree
- F(x, F(y,z)) = F(F(x,y), z) (associativity).
The simplest example is the additive formal group law F(x, y) = x + y.
The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
More generally, an n-dimensional formal group law is a collection of n power series
Fi(x1, x2, ..., xn, y1, y2, ..., yn) in 2n variables, such that
- F(x,y) = x + y + terms of higher degree
- F(x, F(y,z)) = F(F(x,y), z)
where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on.
The formal group law is called commutative if F(x,y) = F(y,x).
Prop. If R is ZZ-torsion free then and formal group law over R is commutative.
proof. The torsion freeness gives us the exponential and logarithm which allows us to write F as F(x,y) = exp(log(x) + log(y)).
There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0.
A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that
-
- G(f(x), f(y)) = f(F(x, y)).
A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)= x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
Examples
- The additive formal group law is given by
- The multiplicative formal group law is given by
This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b) = ab. If we "change coordinates" to make 0 the identity by putting a = 1 + x, b = 1 + y, and G = 1 + F, then we find that
F(x, y) = x + y + xy.
Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x) − 1. Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.
- More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of dimension n, by taking coordinates at the identity and writing down the formal power series expansion of the product map. The additive and multiplicative formal group laws are obtained in this way from the additive and multiplicative algebraic groups. Another important special case of this is the formal group (law) of an elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
(or abelian varietyAbelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
).
- F(x,y) = (x + y)/(1 + xy) is a formal group law coming from the addition formula for the hyperbolic tangent function: tanh(x + y) = F(tanh(x), tanh(y)), and is also the formula for addition of velocities in special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
(with the speed of light equal to 1).
is a formal group law over Z[1/2] found by Euler, in the form of the addition formula for an elliptic integral:
Lie algebras
Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F2 of the formal group law.- [x,y] = F2(x,y) − F2(y,x)
The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group:
-
- Lie groups → Formal group laws → Lie algebras
Over fields of characteristic 0, formal group laws are essentially the same as finite dimensional Lie algebras: more precisely, the functor from finite dimensional formal group laws to finite dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p>0.
The logarithm of a commutative formal group law
If F is a commutative n-dimensional formal group law over a commutative Q-algebra R, then it is strictly isomorphic to the additive formal group law. In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that-
- f(F(x,y)) = f(x) + f(y)
Examples:
- The logarithm of F(x, y) = x + y is f(x) = x.
- The logarithm of F(x, y) = x + y + xy is f(x) = log(1 + x), because log(1 + x + y + xy) = log(1 + x) + log(1 + y).
If R does not contain the rationals, a map f can be constructed by extension of scalars to R⊗Q, but this will send everything to zero if R has positive characteristic. Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R⊗Q, and then proving that the coefficients of the corresponding formal group over R⊗Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W(R) of Witt vectors, and reduces to R at the end.
The formal group ring of a formal group law
The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of a group and to the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups.
For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes messier.
Suppose that F is a (1-dimensional) formal group law over R. Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
H constructed as follows.
- As an R-module, H is free with a basis 1=D(0), D(1), D(2), ...
- The coproduct Δ is given by ΔD(n) = ∑D(i) ⊗ D(n−i) (so the dual of this coalgebra is just the ring of formal power series).
- The counit η is given by the coefficient of D(0).
- The identity is 1 = D(0).
- The antipode S takes D(n) to (−1)nD(n).
- The coefficient of D(1) in the product D(i)D(j) is the coefficient of xiyj in F(x, y).
Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.
Formal group laws as functors
Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is Nn where N is the set of nilpotentNilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.
This makes F into a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from commutative R-algebras S to groups.
We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups. For example, this allows us to define F(Zp) with values in the p-adic numbers.
The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and ηg = 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form
- D(0) + D(1)x + D(2)x2 + ...
for nilpotent elements x. In particular we can identify the group-like elements of H⊗S with the nilpotent elements of S, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S).
The height of a formal group law
Suppose that f is a homomorphism between one dimensional formal group laws over a field of characteristic p > 0. Then f is either zero, or the first nonzero term in its power series expansion is for some non-negative integer h, called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞.The height of a one dimensional formal group law over a field of characteristic p > 0 is defined to be the height of its multiplication by p map.
Two one dimensional formal group laws over an algebraically closed field of characteristic p > 0 are isomorphic if and only if they have the same height, and the height can be any positive integer or ∞.
Examples:
- The additive formal group law F(x, y) = x + y has height ∞, as its pth power map is 0.
- The multiplicative formal group law F(x, y) = x + y + xy has height 1, as its pth power map is (1 + x)p − 1 = xp.
- The formal group law of an elliptic curve has height either one or two, depending on whether the curve is ordinary or supersingular. Supersingularity can be detected by the vanishing of the Eisenstein series
.
Lazard ring
There is a universal commutative one dimensional formal group law over a universal commutative ring defined as follows. We let- F(x, y)
be
- x + y + Σci,j xiyj
for indeterminates
- ci,j,
and we define the universal
ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:
- For any commutative ring S, one dimensional formal group laws over S correspond to ring homomorphisms from R to S.
The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, ... (where ci,j has degree 2(i + j − 1)). Daniel Quillen proved that the coefficient ring of complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
is naturally isomorphic as a graded ring to Lazard's universal ring, explaining the unusual grading.
Formal groups
A formal group is a group objectGroup object
In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets...
in the category of formal scheme
Formal scheme
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme...
s.
- If
is a functor from ArtinianArtinianIn mathematics, Artinian, named for Emil Artin, is an adjective that describes objects that satisfy particular cases of the descending chain condition.*A ring is an Artinian ring if it satisfies the descending chain condition on ideals...
algebras to groups which is left exact, then it representable (G is the functor of points of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits). - If
is a group schemeGroup schemeIn mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
then
, the formal completion of G at the identity has the structure of a formal group. - A smooth group scheme is isomorphic to
. Some people call a formal group scheme smooth if the converse holds. - formal smoothness asserts the existence of lifts of deformations and can apply to formal schemes that are larger than points. A smooth formal group scheme is a special case of a formal group scheme.
- Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections.
- The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group.
Formal groups and formal group laws can also be defined over arbitrary schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object.
The moduli space of formal group laws is a disjoint union of infinite dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite dimensional groupoid of coordinate changes.
Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations. For example, the Serre-Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations.
A formal group is sometimes defined as a cocommutative Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
(usually with some extra conditions added, such as being pointed or connected). This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring.
Some authors use the term formal group to mean formal group law.
Lubin–Tate formal group laws
We let Zp be the ring of p-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other wordsMore generally we can allow e to be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p. All the group laws for different choices of e satisfying these conditions are strictly isomorphic.
For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law.
There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field.
This construction was introduced by , in a successful effort to isolate the local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
part of the classical theory of complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s. It is also a major ingredient in some approaches to local class field theory (e.g., Michiel Hazewinkel,
Local class field theory is easy.
Advances in Math. 18 (1975), no. 2, 148–181).