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).

Properties

Automorphic L-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function L(s,π,r) should be a product over the places v of F of local L functions.

L(s,π,r) = Π L(s,π_v,r_v)

It should have an analytic continuation as a meromorphic function of all complex s, and satisfy a functional equation

L(s,π,r) = ε(s,π,r)L(1–s,π,r^∨)

where the factor ε(s,π,r) is a product of "local constants"

ε(s,π,r) = Π ε(s,π_v,r_v, ψ_v)

almost all of which are 1.

General linear groups

Godement & Jacquet (1972) constructed the automorphic L-functions for general linear groups with r the standard representation and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis.

The Langlands functoriality conjectures imply that all automorphic L-functions are equal to L-functions of general linear groups, so this would prove the analytic continuation and functional equation for them.

References

• Arthur, James; Gelbart, Stephen (1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.), L-functions and arithmetic (Durham, 1989) (PDF), 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
• Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lecture Notes in Mathematics, vol. 260, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070263, ISBN 978-3-540-05797-0, MR0342495
• 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