Automorphic L-function

In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comlplex representation r of the Langlands dual group ^LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by Langlands (1967, 1970, 1971).

References

• Arthur, James; Gelbart, Stephen (1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.), L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge University Press, pp. 1–59, doi:10.1017/CBO9780511526053.003, ISBN 978-0-521-38619-7, MR1110389
• Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR546608
• Langlands, Robert (1967), Letter to Prof. Weil
• Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, vol. 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR0302614
• Langlands, Robert P. (1971) [1967], Euler products, Yale University Press, ISBN 978-0-300-01395-5, MR0419366