This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

General Topology-I

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Infinite Abelian Groups

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Classical Descriptive Set Theory

On computable numbers

We say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space [0, 1] of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of ℝ n , and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.

Computably Isometric Spaces

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

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Foundations of online structure theory

In the last few decades there has been increasing interest in computable model theory. Computable model theory uses the tools of computability theory to explores algorithmic content (e¤ectiveness) of notions, theorems, and constructions in various areas of ordinary mathematics. In algebra this investigation dates back to van der Waerden who in his 1930 book Modern Algebra dened an explicitly given eld as one the elements of which are uniquely represented by distinguishable symbols with which we can perform the eld operations algorithmically. In his pioneering paper on non-factorability of polynomials from 1930, van der Waerden essentially proved that an explicit eld (F;+; ) does not necessarily have an algorithm for splitting polynomials in F [x] into their irreducible factors. Gödels incompleteness theorem from 1931 is an astonishing early result of computable model theory. Gödel showed that there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms.The work of Turing, Gödel, Kleene, Church, Post, and others in the mid-1930s established the rigorous mathematical foundations for the computability theory. In the 1950s, Fröhlich and Shepherdson used the precise notion of a computable function to obtain a collection of results and examples about explicit rings and elds. For example, Fröhlich and

Computable model theory

Algebraic structures computable without delay

Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.

Alexander G. Melnikov

Papers

Computably Isometric Spaces

Foundations of online structure theory

Computable model theory

Algebraic structures computable without delay

Computable functors and effective interpretability