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 L G of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by . and gave surveys of automorphic L-functions.

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 L G of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by . and 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, \pi, r) should be a product over the places v of F of local L functions. L(s, \pi, r) = \prod_v L(s, \pi_v, r_v) Here the automorphic representation \pi = \otimes\pi_v is a tensor product of the representations \pi_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, \pi, r) = \epsilon(s, \pi, r) L(1 - s, \pi, r^\lor) where the factor \epsilon(s, \pi, r) is a product of "local constants" \epsilon(s, \pi, r) = \prod_v \epsilon(s, \pi_v, r_v, \psi_v) almost all of which are 1. ==General linear groups== 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. ==References==

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