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[?].