Tate’s Thesis

Tate’s thesis, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions (Princeton, 1950) first appeared in print as Chapter XV of the conference proceedings Algebraic Number Theory, edited by Cassels and Frölich (Thompson Book Company, Washington, DC, 1967). In it, Tate provides an elegant and unified treatment of the analytic continuation and functional equation of the L-functions attached by Hecke to his Größencharaktere in his pair of papers [7]. The power of the methods of abelian harmonic analysis in the setting of Chevalley’s adèles/idèles provided a remarkable advance over the classical techniques used by Hecke.1 A sketch of the analogous interpretation of classical automorphic forms in terms of nonabelian harmonic analysis—automorphic representations—is given in the next chapter. The development of the theory of automorphic representations for arbitrary reductive groups G is one of the major achievements of mathematics of the later part of the twentieth century. And, of course, this development is still under way, as is evident from the later chapters [6]. In hindsight, Tate’s work may be viewed as giving the theory of automorphic representations and L-functions of the simplest connected reductive group G = GL(1), and so it remains a fundamental reference and starting point for anyone interested in the modern theory of automorphic representations.