##
**Formal groups and applications.**
*(English)*
Zbl 0454.14020

If one opens “Formal Groups and Applications”, one realizes immediately that it has an enthusiastic author who wants to bring the subject across to the reader. The book begins with a table of contents, has a preface, a “Leitfaden and Indicien” and an introduction. After reading all this, one knows that further aids are in the extensive bibliography (of 511 titles), a detailed index and an alphabetical collection of all used notations, subdivided into “standard notations”, “generic notations”, and “incidental notations”. On top of that, each of the seven chapters has its own bibliographical, historical and other notes. Furthermore certain sidelines of the theory appear in extensive remarks within the chapters. The most helpful achievement is – for the reviewer’s taste – that each chapter has a first paragraph of a dozen pages with the title “Definitions and survey of the results”, after which one knows precisely the mathematical objects, their problems, the important properties, theorems, and examples, which will be worked out in the rest of the chapter. No doubt, this book will not only be the standard reference for formal groups but also an excellent textbook for everybody who wants to learn more about formal groups.

The book studies formal groups in the sense of power series, bialgebras are only introduced in the last chapter, the functorial approach is completely omitted.

Chapter I: This chapter gives definitions of the fundamental concepts for one-dimensional formal groups and the most important theorems about them. A one-dimensional formal group law over a ring \(A\) is a formal power series \(F(X,Y)\in A[[X,Y]]\) of the form \(F(X,Y) = X+Y+\) terms of higher degree in \(X\) and \(Y\) such that \(F(X,F(Y,Z)) = F(F(X,Y),Z)\). It is commutative, if \(F(X,Y) = F(Y,X)\). There exists always an inverse \(\iota(X)\in A[[X]]\) such that \(F(X,\iota(X)) = 0\). A homomorphism \(F(X,Y)\rightarrow G(X,Y)\) between two formal group laws over \(A\) is a power series \(\alpha(X)\in A[[X]]^+\) without constant term, such that \(\alpha(F(X,Y)) = G(\alpha(X),\alpha(Y))\). Isomorphisms are homomorphisms which possess an inverse homomorphism.

The fundamental example of an isomorphism consists of the additive (formal) group (law) \(\widehat G_a = X + Y\), the multiplicative group \(\widehat G_m(X,Y)= X + Y + XY\) and the isomorphisms \(E(X) = \displaystyle \sum_{n=1}^\infty \frac{X^n}{n!}\) resp. \(\log (1 + x) = \displaystyle \sum_{n=1}^\infty (-1)^{n+1}\frac{X^n}{n}\) over the ring \(A = \mathbb Q\) and also over every \(\mathbb Q\)-algebra \(A\). For fields \(A\) of characteristic \(p\), however, \(\widehat G_a\) and \(\widehat G_m\) are not isomorphic.

The first important theorem is, that every one dimensional, commutative formal group law \(G\) (not only \(\widehat G_m)\) over a \(\mathbb Q\)-algebra \(A\) is isomorphic to \(\widehat G_a\) by a generalized logisomorphism \(f(X) \in A[[X]^+\). Thus \(G(X,Y) = f^{-1}(f(X) + f(Y))\) and every \(f\) induces a group law over \(A\). If \(A\) is a \(\mathbb Z\)-algebra, \(A\rightarrow A\otimes \mathbb Q\) injective, and \(f(X)\in A\otimes \mathbb Q[[x]]^+\) satisfies a certain functional equation, then \(G(X,Y) = f^{-1}(f(X) + f(Y))\) has coefficients in \(A\). So \(f\) defines a commutative formal group law over \(A\), which is not necessarily isomorphic to \(\widehat G_a\).

Another important theorem states, that all one-dimensional formal group laws over \(A\) are commutative, iff \(A\) contains no nonzero nilpotent torsion elements.

The main aim of this chapter is, however, to construct a universal commutative one-dimensional formal group law . It consists of a formal group law \(F_U(X,Y)\) over a ring \(L\), such that for every one-dimensional commutative formal group law \(G(X,Y)\) over a ring \(A\) there is a unique ring homomorphism \(\varphi\colon L\rightarrow A\) such that \(\varphi_*F_U(X,Y) = G(X,Y)\). It turns out that \(L = \mathbb Z[U_2,U_3,\ldots]\) is a polynomial ring. Thus every one-dimensional commutative formal group law is of the form \(\varphi_*F_U(X,Y)\) and every ring homomorphism \(\varphi\colon \mathbb Z[U_2,U_3,\ldots] \rightarrow A\) induces such a law. Observe that not all group laws over \(L\) are isomorphic, nor that every group law isomorphic to \(F_U(X,Y)\) over \(L\) is universal. The central technique is given by the functional equation lemma here proved in a very general and powerful form, which is too involved to be quoted here. This is the first time that this lemma is published in such a general form. Examples of formal groups are given with constructions due to Honda resp. Lubin-Tate.

Chapter II: Here one finds the fundamental theory of higher dimensional formal group laws \(F(X,Y)\). The difference to the one-dimensional formal group laws is, that one needs \(n\) formal power series in \(2n\) variables for the \(n\)-dimensional case. The additive group is then \(\widehat G_a^n\), the multiplicative group \(\widehat G_m\) (the automorphism group of a 1-dimensional space) changes to a noncommutative \(n^2\)-dimensional general linear group. Part of the 1-dimensional theory can be extended, in particular, the functional equation lemma. As one important consequence one obtains, that every commutative formal group law over a \(\mathbb Q\)-algebra \(A\) is isomorphic to (a power of) the additive group \(\widehat G_a^n\). Again there is a universal \(n\)-dimensional formal group law with corresponding ring \(\mathbb Z[U_1,U_2,\ldots]\). But here the differences from the 1-dimensional case begin. It turns out that this universal formal group law does not generalize the 1-dimensional case. Curvilinearity, i.e. the vanishing of certain mixed terms in \(F(X,Y)\), is one of the new phenomena here, although every commutative \(n\)-dimensional formal group law is isomorphic to a curvilinear one.

After the construction of generalized Honda and Lubin-Tate group laws, the characteristic zero Lie theory is done. Here the main theorem is, that the category of finite-dimensional formal group laws and the category of (free) finite rank Lie algebras are equivalent.

Chapter III studies groups of curves in commutative formal group laws \(F(X,Y)\). The elements are \(n\)-tuples of power series \(\gamma(t)\in A[[t]]^+\). The “addition” is \(\gamma_1(t) + \gamma_2(t) = F(\gamma_1(t), \gamma_2(t))\). The curves in \(F(X,Y)\) form a group with additional operators, the Verschiebung, Frobenius and homothety operators. The Frobenius operators are introduced by primitive roots of unity as well as by elementary symmetric polynomials. Curves in \(F(X,Y)\) are called \(p\)-typical (for a prime \(p)\) if the \(q\)-th Frobenius operators for all primes \(q\ne p\) vanish on them. \(F(X,Y)\) is \(p\)-typical if the curves which consist just of \(p^i\)-th powers of \(t\) are \(p\)-typical (in characteristic zero). Again there is a universal \(p\)-typical group law. Over \(\mathbb Z_{(p)}\) it turns out that every formal group law is isomorphic to a \(p\)-typical formal group law.

The main aim of this chapter is the introduction of generalized Witt polynomials using the coefficients of the logarithm of a one-dimensional formal group law \(F(X,Y)\) in characteristic zero. Using these Witt polynomials, rings (groups) of infinite Witt vectors and an Artin-Hasse exponential mapping over \(F(X,Y)\) are constructed and studied. Special cases are the ordinary infinite Witt vectors coming from \(\widehat G_m^-(X,Y) =X +Y - XY\), the Witt vectors with respect to a prime \(p\) and “ramified Witt vectors” over certain Lubin-Tate formal groups.

In chapter IV homomorphisms and isomorphisms of commutative formal group laws are studied in more detail. In particular cases a classification of all isomorphisms by Eisenstein polynomials is given. Furthermore in characteristic \(p\) the concept of height of a homomorphism is introduced, the height of a formal group law corresponds to the height of the endomorphism induced by \(p\). The concept of height allows the definition of a topology on the set of homomorphisms between certain formal group laws, such that this set is complete. This leads to the classification of the one-dimensional formal group laws by their heights and Galois descent. Also the moduli problem is attacked, the problem of classifying liftings of one-dimensional formal group laws from a residue field to a complete noetherian ring. A large part of this chapter is filled by the theory of formal \(A\)-modules, i.e. formal group laws on which an algebra \(A\) operates by endomorphisms. It turns out that most of the theory developed up to here, especially in the one-dimensional case, generalizes to formal \(A\)-modules.

In chapter V it is shown that each commutative formal group law defines a set of curves. These curves form an abelian complete Hausdorff topological group with certain continuous operators, a so-called reduced Cartier-Dieudonné module. This construction is a functorial equivalence, and the functor from commutative formal group laws to reduced Cartier-Dieudonné modules is representable by the formal group law of Witt vectors. Similar results hold for \(A\)-modules. Over an algebraically closed field, the finite dimensional \(A\)-modules may be decomposed “up to isogeny” into very simple components. A paragraph on the “tapis de Cartier” closes this chapter. Chapter VI introduces various applications of formal group laws, to name a few: complex oriented cohomology theories, Brown-Peterson cohomology, Tate modules, local class field theory, and zeta functions of elliptic curves.

The last, fairly short, chapter VII gives an introduction on how to study for formal group laws their co- and contravariant bialgebras (or hyperalgebras), their connection with the Lie algebra of a formal group law, and how curves can be considered as sequences of divided powers. Furthermore, a noncommutative analogue of the ring of Witt vectors is constructed.

In two appendices a brief introduction to power series rings is given for the convenience of the reader and some more examples where the whole theory might be applied, e.g. global class field theory, \(p\)-divisible groups, lifting abelian varieties, arithmetic algebraic geometry, \(L\)-functions on elliptic curves, extraordinary \(K\)-theories, and Hopf rings.

The book studies formal groups in the sense of power series, bialgebras are only introduced in the last chapter, the functorial approach is completely omitted.

Chapter I: This chapter gives definitions of the fundamental concepts for one-dimensional formal groups and the most important theorems about them. A one-dimensional formal group law over a ring \(A\) is a formal power series \(F(X,Y)\in A[[X,Y]]\) of the form \(F(X,Y) = X+Y+\) terms of higher degree in \(X\) and \(Y\) such that \(F(X,F(Y,Z)) = F(F(X,Y),Z)\). It is commutative, if \(F(X,Y) = F(Y,X)\). There exists always an inverse \(\iota(X)\in A[[X]]\) such that \(F(X,\iota(X)) = 0\). A homomorphism \(F(X,Y)\rightarrow G(X,Y)\) between two formal group laws over \(A\) is a power series \(\alpha(X)\in A[[X]]^+\) without constant term, such that \(\alpha(F(X,Y)) = G(\alpha(X),\alpha(Y))\). Isomorphisms are homomorphisms which possess an inverse homomorphism.

The fundamental example of an isomorphism consists of the additive (formal) group (law) \(\widehat G_a = X + Y\), the multiplicative group \(\widehat G_m(X,Y)= X + Y + XY\) and the isomorphisms \(E(X) = \displaystyle \sum_{n=1}^\infty \frac{X^n}{n!}\) resp. \(\log (1 + x) = \displaystyle \sum_{n=1}^\infty (-1)^{n+1}\frac{X^n}{n}\) over the ring \(A = \mathbb Q\) and also over every \(\mathbb Q\)-algebra \(A\). For fields \(A\) of characteristic \(p\), however, \(\widehat G_a\) and \(\widehat G_m\) are not isomorphic.

The first important theorem is, that every one dimensional, commutative formal group law \(G\) (not only \(\widehat G_m)\) over a \(\mathbb Q\)-algebra \(A\) is isomorphic to \(\widehat G_a\) by a generalized logisomorphism \(f(X) \in A[[X]^+\). Thus \(G(X,Y) = f^{-1}(f(X) + f(Y))\) and every \(f\) induces a group law over \(A\). If \(A\) is a \(\mathbb Z\)-algebra, \(A\rightarrow A\otimes \mathbb Q\) injective, and \(f(X)\in A\otimes \mathbb Q[[x]]^+\) satisfies a certain functional equation, then \(G(X,Y) = f^{-1}(f(X) + f(Y))\) has coefficients in \(A\). So \(f\) defines a commutative formal group law over \(A\), which is not necessarily isomorphic to \(\widehat G_a\).

Another important theorem states, that all one-dimensional formal group laws over \(A\) are commutative, iff \(A\) contains no nonzero nilpotent torsion elements.

The main aim of this chapter is, however, to construct a universal commutative one-dimensional formal group law . It consists of a formal group law \(F_U(X,Y)\) over a ring \(L\), such that for every one-dimensional commutative formal group law \(G(X,Y)\) over a ring \(A\) there is a unique ring homomorphism \(\varphi\colon L\rightarrow A\) such that \(\varphi_*F_U(X,Y) = G(X,Y)\). It turns out that \(L = \mathbb Z[U_2,U_3,\ldots]\) is a polynomial ring. Thus every one-dimensional commutative formal group law is of the form \(\varphi_*F_U(X,Y)\) and every ring homomorphism \(\varphi\colon \mathbb Z[U_2,U_3,\ldots] \rightarrow A\) induces such a law. Observe that not all group laws over \(L\) are isomorphic, nor that every group law isomorphic to \(F_U(X,Y)\) over \(L\) is universal. The central technique is given by the functional equation lemma here proved in a very general and powerful form, which is too involved to be quoted here. This is the first time that this lemma is published in such a general form. Examples of formal groups are given with constructions due to Honda resp. Lubin-Tate.

Chapter II: Here one finds the fundamental theory of higher dimensional formal group laws \(F(X,Y)\). The difference to the one-dimensional formal group laws is, that one needs \(n\) formal power series in \(2n\) variables for the \(n\)-dimensional case. The additive group is then \(\widehat G_a^n\), the multiplicative group \(\widehat G_m\) (the automorphism group of a 1-dimensional space) changes to a noncommutative \(n^2\)-dimensional general linear group. Part of the 1-dimensional theory can be extended, in particular, the functional equation lemma. As one important consequence one obtains, that every commutative formal group law over a \(\mathbb Q\)-algebra \(A\) is isomorphic to (a power of) the additive group \(\widehat G_a^n\). Again there is a universal \(n\)-dimensional formal group law with corresponding ring \(\mathbb Z[U_1,U_2,\ldots]\). But here the differences from the 1-dimensional case begin. It turns out that this universal formal group law does not generalize the 1-dimensional case. Curvilinearity, i.e. the vanishing of certain mixed terms in \(F(X,Y)\), is one of the new phenomena here, although every commutative \(n\)-dimensional formal group law is isomorphic to a curvilinear one.

After the construction of generalized Honda and Lubin-Tate group laws, the characteristic zero Lie theory is done. Here the main theorem is, that the category of finite-dimensional formal group laws and the category of (free) finite rank Lie algebras are equivalent.

Chapter III studies groups of curves in commutative formal group laws \(F(X,Y)\). The elements are \(n\)-tuples of power series \(\gamma(t)\in A[[t]]^+\). The “addition” is \(\gamma_1(t) + \gamma_2(t) = F(\gamma_1(t), \gamma_2(t))\). The curves in \(F(X,Y)\) form a group with additional operators, the Verschiebung, Frobenius and homothety operators. The Frobenius operators are introduced by primitive roots of unity as well as by elementary symmetric polynomials. Curves in \(F(X,Y)\) are called \(p\)-typical (for a prime \(p)\) if the \(q\)-th Frobenius operators for all primes \(q\ne p\) vanish on them. \(F(X,Y)\) is \(p\)-typical if the curves which consist just of \(p^i\)-th powers of \(t\) are \(p\)-typical (in characteristic zero). Again there is a universal \(p\)-typical group law. Over \(\mathbb Z_{(p)}\) it turns out that every formal group law is isomorphic to a \(p\)-typical formal group law.

The main aim of this chapter is the introduction of generalized Witt polynomials using the coefficients of the logarithm of a one-dimensional formal group law \(F(X,Y)\) in characteristic zero. Using these Witt polynomials, rings (groups) of infinite Witt vectors and an Artin-Hasse exponential mapping over \(F(X,Y)\) are constructed and studied. Special cases are the ordinary infinite Witt vectors coming from \(\widehat G_m^-(X,Y) =X +Y - XY\), the Witt vectors with respect to a prime \(p\) and “ramified Witt vectors” over certain Lubin-Tate formal groups.

In chapter IV homomorphisms and isomorphisms of commutative formal group laws are studied in more detail. In particular cases a classification of all isomorphisms by Eisenstein polynomials is given. Furthermore in characteristic \(p\) the concept of height of a homomorphism is introduced, the height of a formal group law corresponds to the height of the endomorphism induced by \(p\). The concept of height allows the definition of a topology on the set of homomorphisms between certain formal group laws, such that this set is complete. This leads to the classification of the one-dimensional formal group laws by their heights and Galois descent. Also the moduli problem is attacked, the problem of classifying liftings of one-dimensional formal group laws from a residue field to a complete noetherian ring. A large part of this chapter is filled by the theory of formal \(A\)-modules, i.e. formal group laws on which an algebra \(A\) operates by endomorphisms. It turns out that most of the theory developed up to here, especially in the one-dimensional case, generalizes to formal \(A\)-modules.

In chapter V it is shown that each commutative formal group law defines a set of curves. These curves form an abelian complete Hausdorff topological group with certain continuous operators, a so-called reduced Cartier-Dieudonné module. This construction is a functorial equivalence, and the functor from commutative formal group laws to reduced Cartier-Dieudonné modules is representable by the formal group law of Witt vectors. Similar results hold for \(A\)-modules. Over an algebraically closed field, the finite dimensional \(A\)-modules may be decomposed “up to isogeny” into very simple components. A paragraph on the “tapis de Cartier” closes this chapter. Chapter VI introduces various applications of formal group laws, to name a few: complex oriented cohomology theories, Brown-Peterson cohomology, Tate modules, local class field theory, and zeta functions of elliptic curves.

The last, fairly short, chapter VII gives an introduction on how to study for formal group laws their co- and contravariant bialgebras (or hyperalgebras), their connection with the Lie algebra of a formal group law, and how curves can be considered as sequences of divided powers. Furthermore, a noncommutative analogue of the ring of Witt vectors is constructed.

In two appendices a brief introduction to power series rings is given for the convenience of the reader and some more examples where the whole theory might be applied, e.g. global class field theory, \(p\)-divisible groups, lifting abelian varieties, arithmetic algebraic geometry, \(L\)-functions on elliptic curves, extraordinary \(K\)-theories, and Hopf rings.

Reviewer: Bodo Pareigis (München)

### MSC:

14L05 | Formal groups, \(p\)-divisible groups |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14L17 | Affine algebraic groups, hyperalgebra constructions |

13F35 | Witt vectors and related rings |

13F25 | Formal power series rings |