Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexgave _[[automorphic L-function]]_ a minimum of an Idea-section, presently it reads as follows: *** An _automorphic L-function_ $L_\pi$ is an [[L-function]] built from an [[automorphic representation]] $\pi$, in nonabelian generalization of how a [[Dirichlet L-function]] $L_\chi$ is associated to a [[Dirichlet character]] $\chi$ (which is an automorphic form on the (abelian) [[idele group]]). In analogy to how [[Artin reciprocity]] implies that to every 1-dimensional [[Galois representation]] $\sigma$ there is a [[Dirichlet character]] $\chi$ such that the [[Artin L-function]] $L_\sigma$ equals the [[Dirichlet L-function]] $L_\chi$, so the [[conjecture|conjectured]] [[Langlands correspondence]] says that to every $n$-dimensional [[Galois representation]] $\sigma$ there is an [[automorphic representation]] $\pi$ such that the automorphic L-function $L_\pi$ equals the [[Artin L-function]] $L_\sigma$. ***

   gave automorphic L-function a minimum of an Idea-section, presently it reads as follows:

   ---

   An automorphic L-function $L_\pi$ is an L-function built from an automorphic representation $\pi$, in nonabelian generalization of how a Dirichlet L-function $L_\chi$ is associated to a Dirichlet character $\chi$ (which is an automorphic form on the (abelian) idele group).

   In analogy to how Artin reciprocity implies that to every 1-dimensional Galois representation $\sigma$ there is a Dirichlet character $\chi$ such that the Artin L-function $L_\sigma$ equals the Dirichlet L-function $L_\chi$, so the conjectured Langlands correspondence says that to every $n$-dimensional Galois representation $\sigma$ there is an automorphic representation $\pi$ such that the automorphic L-function $L_\pi$ equals the Artin L-function $L_\sigma$.

   ---

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 17th 2015
   • (edited Jul 17th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded at the beginning of the [Idea section](http://ncatlab.org/nlab/show/automorphic+L-function#Idea) briefly the conceptual statement that passing from an automorphic form to its automorphic L-function is the generalization of the Mellin transform taking an automorphic form to its zeta function.

   added at the beginning of the Idea section briefly the conceptual statement that passing from an automorphic form to its automorphic L-function is the generalization of the Mellin transform taking an automorphic form to its zeta function.