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 (1991) gave surveys of automorphic L-functions.

Automorphic L {\displaystyle 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 ) {\displaystyle L(s,\pi ,r)} should be a product over the places v {\displaystyle v} of F {\displaystyle F} of local L {\displaystyle L} functions.

L ( s , π , r ) = ∏ v L ( s , π v , r v ) {\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}

Here the automorphic representation π = ⊗ π v {\displaystyle \pi =\otimes \pi _{v}} is a tensor product of the representations π v {\displaystyle \pi _{v}} of local groups.

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

L ( s , π , r ) = ϵ ( s , π , r ) L ( 1 − s , π , r ∨ ) {\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}

where the factor ϵ ( s , π , r ) {\displaystyle \epsilon (s,\pi ,r)} is a product of "local constants"

ϵ ( s , π , r ) = ∏ v ϵ ( s , π v , r v , ψ v ) {\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}

almost all of which are 1.

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. 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.

• Grand Riemann hypothesis

• 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, MR 1110389
• 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, vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, doi:10.1090/pspum/033.2/546608, ISBN 978-0-8218-1437-6, MR 0546608
• Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004), Lectures on automorphic L-functions, Fields Institute Monographs, vol. 20, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3516-6, MR 2071722
• Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen (1987), Explicit Constructions of Automorphic L-Functions, Lecture Notes in Mathematics, vol. 1254, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0078125, ISBN 978-3-540-17848-4, MR 0892097
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