**Tate's thesis**, on the analytic properties of the class of L-function introduced by Erich Hecke, is one of the relatively few such dissertations that have become a by-word. In it the methods, novel for that time, of Fourier analysis on groups of *adeles*, were worked out to recover Hecke's results.

Subsequently Tate worked with Emil Artin to give a treatment of class field theory based on cohomology of groups, explaining the content as the Galois cohomology of idele classes. In the following decades Tate extended the reach of Galois cohomology: duality, abelian varieties, the Tate-Shafarevich group, and relations with algebraic K-theory.

He made a number of individual and important contributions to p-adic theory: the Lubin-Tate local theory of complex multiplication of formal groups; rigid analytic spaces; the 'Tate curve' parametrisation for p-adic elliptic curves; p-divisible (Tate-Barsotti) groups. Many of his results were not immediately published and were written up by Jean-Pierre Serre. They collaborated on a major published paper on abelian varieties.

The **Tate conjectures** are the equivalent for etale cohomology of the Hodge conjecture. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.