The Work of John Tate

CLASS FIELD THEORY: CLASS FORMATIONS Tate’s theorem (see (4) above) shows that, in order to have a class field theory over a field k, all one needs is, for each system of fields ksep L K k; ŒLWk <1; L=K Galois, a G.L=K/-module CL and a “fundamental class” uL=K 2 H .G.L=K/;CL/ satisfying Tate’s hypotheses; the pairs .CL;uL=K/ should also satisfy certain natural conditions when K and L vary. Then Tate’s theorem then provides “reciprocity” isomorphisms C L ' !G=ŒG;G; G DG.L=K/; 1 HECKE L-SERIES AND THE COHOMOLOGY OF NUMBER FIELDS 11 Artin and Tate (1961, Chapter 14) formalized this by introducing the abstract notion of a class formation. For example, for any nonarchimedean local field k, there is a class formation with CL D L for any finite extension L of k, and for any global field, there is a class formation with CL D JL=L . In both cases, uL=K is the fundamental class. Let k be an algebraic function field in one variable with algebraically closed constant field. Kawada and Tate (1955a) show that there is a class formation for unramified extensions ofK with CL the dual of the group of divisor classes of L. In this way they obtain a “pseudo class field theory” for k, which they examine in some detail when k D C.