John Tate

Class field theory

This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians. In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new footnotes, additional references, and historical comments.

p-Divisible Groups

After a brief review of facts about finite locally free commutative group schemes in § 1, we define p-divisible groups in § 2, and discuss their relation to formal Lie groups. The § 3 contains some theorems about the action of Gal (K/K) on the completion C of the algebraic closure of a local field K of characteristic 0. In § 4 these theorems are applied to obtain information about the Galois module of points of finite order on a p-divisible group G defined over the ring of integers R in such a field K, and to prove that G is determined by that Galois module, or, what is the same, by its generic fiber G ×R K.

Some Algebras Associated to Automorphisms of Elliptic Curves

The main object of this paper is to relate a certain type of graded algebra, namely the regular algebras of dimension 3, to automorphisms of elliptic curves. Some of the results were announced in [V]. A graded algebra A is called regular if it has finite global dimension, polynomial growth, and is Gorenstein. The precise definitions are reviewed in Section 2. As was shown in [A-S], there are two basic possibilities for a regular algebra A of (global) dimension 3 which is generated in degree 1. Either A can be presented by 3 generators and 3 quadratic relations, or else by 2 generators and 2 cubic relations. Throughout this paper, A will denote an algebra so presented, over a ground field k.

Formal Complex Multiplication in Local Fields

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On the Characters of Finite Groups

where each A is an irreducible character of some elementary subgroup (si of (M, ip* designates the character of (M induced by AXv , and where the ai belong to the ring Z of rational integers. Here, an elementary group is defined as a group which is the direct product of a cyclic group and a p-group for some prime number p. By a generalized character of 5, we shall mean the difference of two characters of 5. Thus, if xI, X2, * * , Xk are the irreducible characters of 5, the generalized characters have the form

Finite Flat Group Schemes

The kernel of an isogeny of degree n of abelian varieties of dimension g is, at a place of good reduction, a finite flat group scheme of order n2g over the local ring of the place. That is perhaps the main reason for studying finite flat group schemes, although they are interesting enough in their own right, and it is in any case the reason a discussion of them appears in this volume. For that reason also, the commutative case is the most important for us, and it is in that case that the theory is most interesting and highly developed by far. Nevertheless we do not assume commutativity at the beginning and develop the basics of the theory without that assumption.

The Centers of 3-Dimensional Sklyanin Algebras

Publisher Summary This chapter describes the centers of three-dimensional Sklyanin algebras. In noncommutative situations, the problem of showing that rings are noetherian is much more difficult than in the commutative case. There is no Hilbert Basis Theorem, even in the case of algebras that are deformations of commutative polynomial rings. Of the seven generic types of these algebras, three come from reducible cubics and four from smooth irreducible ones. The four cases correspond to the nature of the automorphism, which can be a translation, a reflection, or a complex multiplication of order 3 or 4. Such algebra is called a Sklyanin algebra, because Sklyanin gave a construction of a four-dimensional analog. The definition in dimension 4 is in fact more subtle than in dimension 3, but the generalization of the construction to higher dimensions, obtaining from a triple is shown. There are a few other nontrivial examples of regular algebras of dimension known, but a complete classification analogous to that in dimension 3, even for quadratic ones seems far away, even in dimension 4.

Geometry and Arithmetic

Everyone knows what a rational number is, a quotient of two integers. We call a point (x, y) in the plane a rational point if both of its coordinates are rational numbers. We call a line a rational line if the equation of the line can be written with rational numbers, that is, if it has an equation

The Group of Rational Points

\displaystyle{ax + by + c = 0}