Editing Functional equation (L-function) (section)

== Theory of functional equations ==

A unified theory of such functional equations was given by [[Erich Hecke]], and the theory was taken up again in [[Tate's thesis]] by [[John Tate (mathematician)|John Tate]]. Hecke found generalised characters of number fields, now called [[Hecke character]]s, for which his proof (based on [[theta function]]s) also worked. These characters and their associated L-functions are now understood to be strictly related to [[complex multiplication]], as the Dirichlet characters are to [[cyclotomic field]]s.

There are also functional equations for the [[local zeta-function]]s, arising at a fundamental level for the (analogue of) [[Poincaré duality]] in [[étale cohomology]]. The Euler products of the [[Hasse–Weil zeta-function]] for an [[algebraic variety]] ''V'' over a number field ''K'', formed by reducing ''modulo'' [[prime ideal]]s to get local zeta-functions, are conjectured to have a ''global'' functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from [[automorphic representation]] theory seems required to get the functional equation. The [[Taniyama–Shimura conjecture]] was a particular case of this as general theory. By relating the gamma-factor aspect to [[Hodge theory]], and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.