In his doctoral thesis, Tate introduced harmonic analysis into number theory, paving the way for the adelic approach to automorphic forms and the Langlands programme.

In his thesis, which has become a classic, he proved the functional equation for Hecke's L-series by a novel method involving Fourier analysis on idele groups.

His work on class field theory and Galois cohomology over local and global fields, especially his duality theory, underpins much of modern number theory; and the Tate cohomology groups for finite groups, which he invented for use in class field theory, are a standard tool of algebraists. Tate's deep insights have had a crucial impact on the development of arithmetic algebraic geometry from the sixties onwards. Perhaps most celebrated are his conjectures about algebraic cycles on varieties over finite and global fields, formulated 35 years ago but still largely unproved. Equally striking is his seminal 1966 paper 'p-divisible groups', which for the first time recognised the richness of p-adic representations of the absolute Galois group of a p-adic field, as well as indicating the existence of a p-adic analogue of Hodge theory. This is now a key tool in understanding the arithmetic of algebraic varieties. Tate's work on classification of abelian varieties over finite fields is a core part of standard theory, underpinning almost all work on the L-functions of Shimura varieties as well as being the starting point for the study of motives over finite fields. Through his discovery of rigid analytic spaces, he established new foundations for p-adic global analysis which have wide applicability in number theory, algebraic geometry and representation theory. The theory of elliptic curves owes an enormous amount to his contributions, both theoretical and computational; the theory of height functions (Neron-Tate and Mazur-Tate) and descent theory (including his construction of the notorious Shafarevich-Tate group) are of key importance in understanding the arithmetic of elliptic curves, and Tate's algorithm for determining the bad reduction of an elliptic curve plays an equally important role in computation. Other contributions of deep significance include his work with Serre on the deformation theory of abelian varieties, his contributions to algebraic K-theory and its relation with Galois cohomology, his work on the Stark conjectures, and most recently his work in non-commutative ring theory.

... for scientific accomplishments spanning four and a half decades. He has been deeply influential in many of the important developments in algebra, algebraic geometry, and number theory during this time.

... for his creation of fundamental concepts in algebraic number theory.

For over a quarter of a century, Professor John Tate's ideas have dominated the development of arithmetic algebraic geometry. Tate has introduced path breaking techniques and concepts, that initiated many theories which are very much alive today. These include Fourier analysis on local fields and adele rings, Galois cohomology, the theory of rigid analytic varieties, and p-divisible groups and p-adic Hodge decompositions, to name but a few. Tate has been an inspiration to all those working on number theory. Numerous notions bear his name: Tate cohomology of a finite group, Tate module of an abelian variety, Tate-Shafarevich group, Lubin-Tate groups, Neron-Tate heights, Tate motives, the Sato-Tate conjecture, Tate twist, Tate elliptic curve, and others. John Tate is a revered name in algebraic number theory.

The authors' goal has been to write a textbook in a technically difficult field which is accessible to the average undergraduate mathematics major, and it seems that they have succeeded admirably. The book is quite delightful. ... The most obvious drawback to a text for undergraduates in a field such as this is that it is not possible to be entirely rigorous, and so, as the authors declare, "much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than rigorously prove." An appendix does develop the necessary algebraic geometry, but throughout the book the approach to the underlying geometry is informal, allowing a more rapid and intuitive access to the number theory.

... for his vast and lasting impact on the theory of numbers.

The theory of numbers stretches from the mysteries of prime numbers to the ways in which we store, transmit, and secure information in modern computers. Over the past century it has developed into one of the most elaborate and sophisticated branches of mathematics, interacting profoundly with other key areas. John Tate is a prime architect of this development. John Tate's scientific accomplishments span six decades. A wealth of essential mathematical ideas and constructions were initiated by Tate and later named after him, such as the Tate module, Tate curve, Tate cycle, Hodge-Tate decompositions, Tate cohomology, Serre-Tate parameter, Lubin-Tate group, Tate trace, Shafarevich-Tate group, Néron-Tate height, to mention just a few. According to the Abel committee, "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics."