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 representation π of a reductive group G over a global field and a finite-dimensional complex 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) gave surveys of automorphic L-functions.

Properties

Automorphic [math]\displaystyle{ L }[/math]-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function [math]\displaystyle{ L(s, \pi, r) }[/math] should be a product over the places [math]\displaystyle{ v }[/math] of [math]\displaystyle{ F }[/math] of local [math]\displaystyle{ L }[/math] functions.

[math]\displaystyle{ L(s, \pi, r) = \prod_v L(s, \pi_v, r_v) }[/math]

Here the automorphic representation [math]\displaystyle{ \pi = \otimes\pi_v }[/math] is a tensor product of the representations [math]\displaystyle{ \pi_v }[/math] of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex [math]\displaystyle{ s }[/math], and satisfy a functional equation

[math]\displaystyle{ L(s, \pi, r) = \epsilon(s, \pi, r) L(1 - s, \pi, r^\lor) }[/math]

where the factor [math]\displaystyle{ \epsilon(s, \pi, r) }[/math] is a product of "local constants"

[math]\displaystyle{ \epsilon(s, \pi, r) = \prod_v \epsilon(s, \pi_v, r_v, \psi_v) }[/math]

almost all of which are 1.

General linear groups

(Godement Jacquet) 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. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.

In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.

See also

• Grand Riemann hypothesis

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, http://www.claymath.org/library/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, doi:10.1090/pspum/033.2/546608, ISBN 978-0-8218-1437-6, https://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, https://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 
• 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 
• Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", Amer. J. Math. 105: 367–464, doi:10.2307/2374264 
• 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, http://publications.ias.edu/rpl/section/21 
• Langlands, Robert P. (1971), Euler products, Yale University Press, ISBN 978-0-300-01395-5, http://publications.ias.edu/rpl/paper/37 
• Shahidi, F. (1981), "On certain "L"-functions", Amer. J. Math. 103: 297–355, doi:10.2307/2374219