Historical remarks

The analytic theory of L-functions associated to modular forms was developed by Hecke (see [Hecke, 1936]), and later extended to non-holomorphic automorphic forms in [Maass, 1946, 1949]. The analytic continuation and functional equation of such an L-function was obtained by taking the Mellin transform of an automorphic form f(z) and applying the modular relation z ↦ -z-1 as in [Riemann, 1859]. This approach was further generalized to the adelic representation theoretic setting in [Gelfand-Graev-Pyatetski-Shapiro, 1969], [Jacquet-Langlands, 1970].

A simple algebra is an algebra A which contains no non-trivial two-sided ideals and for which there exists elements a, a′ ∈ A for which aa′ ≠ 0. It was proved in [Wedderburn, 1907] that every simple algebra of finite degree n is isomorphic to the algebra of n × n matrices with entries in a division ring. Since the foundational papers of [Tate, 1950], [Iwasawa, 1952], on the analytic continuation and functional equation of L-functions for GL(1, Aℚ) (see Section 2.2, 2.3), it has been expected that there should be a generalization to L-functions of simple algebras over ℚ. This was done in [Fujisaki, 1958, 1962] for L-functions with abelian characters. The work of Fujisaki includes earlier constructions of zeta functions in [Hey, 1929] and [Eichler, 1938]. A theory of Iwasawa-Tate type for non-abelian characters was laid out in [Godement, 1958/1959] and extended in [Tamagawa, 1963] who also obtained the theory of the Euler product by showing that the global L-function is a product of local L-functions.