Tate's thesis, on the analytic properties of the class of Lfunction introduced by Erich Hecke[?], is one of the relatively few such dissertations that have become a byword. 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 TateShafarevich group, and relations with algebraic Ktheory.
He made a number of individual and important contributions to padic theory: the LubinTate local theory of complex multiplication of formal groups; rigid analytic spaces; the 'Tate curve' parametrisation for padic elliptic curves; pdivisible (TateBarsotti) groups. Many of his results were not immediately published and were written up by JeanPierre 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 ladic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tatetwisted 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[?].
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