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).

Borel (1979) and Arthur & Gelbart (1991) gave surveys of automorphic L-functions.

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)

Here the automorphic representation π=⊗π_v is a tensor product of the representations π_v of local groups.

The L-function is expected to 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 (so-called standard L-functions) 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., L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153, Cambridge University Press, pp. 1–59, doi:10.1017/CBO9780511526053.003, ISBN 978-0-521-38619-7, MR1110389, http://www.claymath.org/cw/arthur/pdf/automorphic-L.pdf 
• Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR546608, http://www.ams.org/publications/online-books/pspum332-index 
• Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004), Lectures on automorphic L-functions, Fields Institute Monographs, 20, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3516-6, MR2071722, http://books.google.com/books?id=jb3ZCp0-MQsC 
• Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen (1987), Explicit constructions of automorphic L-functions, Lecture Notes in Mathematics, 1254, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0078125, ISBN 978-3-540-17848-4, MR892097 
• Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lecture Notes in Mathematics, 260, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070263, ISBN 978-3-540-05797-0, MR0342495 
• Langlands, Robert (1967), Letter to Prof. Weil, http://publications.ias.edu/rpl/section/21 
• Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR0302614, http://publications.ias.edu/rpl/section/21 
• Langlands, Robert P. (1971) [1967], Euler products, Yale University Press, ISBN 978-0-300-01395-5, MR0419366, http://publications.ias.edu/rpl/paper/37