Langlands program (reviewed)

Langlands theory of automorphic forms ushered in an age of a new understanding of the profundity of L-functions. His small book Euler Products of 1967 contains the beginnings of a general theory.

In 1969 Ogg [MR:0246819] proved the holomorphy and functional equation for the Rankin-Selberg convolution of two inequivalent cusp forms of the same level. This is a degree 4 L-function. In 1972 Jacquet [MR:0562503] considered the Rankin-Selberg convolution of two GL(2) cusp forms and obtained the analytic continuation and functional equation of the associated L-function. In 1979 Winnie Li [MR:0550843] completed the Rankin-Selberg story for two arbitrary GL(2) cusp forms. Her techniques really require the use of Tate's thesis and the theory of automorphic forms as it is virtually impossible to figure out the Euler factors at bad primes without these.

In 1971 Andrianov [MR:0340178] explicitly constructed the L-function for a genus 2 Siegel modular form and gave its analytic continuation and functional equation.

In 1972 Godement and Jacquet [MR:0342495] defined the L-function of a general automorphic form on a reductive group and obtained the analytic continuation and functional equation.

It wasn't until the mid 1970's that Shimura [MR:0382176] and, shortly later but independently, Zagier [MR:0485703] proved that the L-function associated with the symmetric square of a cusp form is entire. See also the footnote in Selberg's paper [MR:1220477] that suggests that Selberg had discovered this many years earlier.

In 1977 Asai [MR:0429751] obtained the holomorphy and functional equation of certain degree 4 L-functions associated with Hilbert modular forms over quadratic fields. These are known as Asai L-functions.

Beginning in 1978 Shahidi [MR:0498494] and completed in 1999 with Kim [MR:1726704] proved that the symmetric cube of a GL(2) L-function is entire and has functional equation.

In 1980 Langlands proved Artin's conjecture for 2-dimensional representations of tetrahedral type (and also for certain octahedral representations) [MR:0574808]. In 1981 Tunnell proved Artin's conjecture for octahedral representations [MR:0621884].

In work beginning in 1981 with [MR:0610479], Shahidi proved the holomorphy and functional equation for many automorphic L-functions.

In 1983 Jacquet, Piatetskii-Shapiro, and Shalika [MR:0701565]obtained the meromorphicity and functional equation for the L-function that is a general Rankin-Selberg convolution of automorphic L-functions.

In 1995 Andrew Wiles [MR:1333035] proved the holomorphy and functional equation of the Hasse-Weil zeta-function of most elliptic curves. In 2001 this work was extended by Breuil, Conrad, Diamond, and Taylor to include all elliptic curves [MR:1839918].