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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2014
   • (edited Aug 28th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave expanded the Idea-section at _[[L-function]]_ in an attempt to transport some actual idea. The main addition are these paragraphs: *** The most canonically defined class of examples of L-functions are the _[[Artin L-functions]]_ defined for any [[Galois representation]] $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the [[Euler products]] of, essentially, [[characteristic polynomials]] of all the [[Frobenius homomorphisms]] acting via $\sigma$. Most other kinds of L-functions are such as to reproduces these [[Artin L-functions]] from more "arithmetic" data: 1. for 1-dimensional [[Galois representations]] $\sigma$ (hence for $n = 1$) [[Artin reciprocity]] produces for each $\sigma$ a [[Dirichlet character]], or more generally a [[Hecke character]] $\chi$, and therefrom is built a _[[Dirichlet L-function]]_ or _[[Hecke L-function]]_ $L_\chi$, respectively, which equals the corresponding [[Artin L-function]] $L_\sigma$; 1. for general $n$-dimensional [[Galois representations]] $\sigma$ the [[conjecture]] of [[Langlands correspondence]] states that there is an [[automorphic representation]] $\pi$ corresponding to $\sigma$ and an _[[automorphic L-function]]_ $L_\pi$ built from that, which equalso the [[Artin L-function]] $L_\sigma$. ***

   have expanded the Idea-section at L-function in an attempt to transport some actual idea. The main addition are these paragraphs:

   ---

   The most canonically defined class of examples of L-functions are the Artin L-functions defined for any Galois representation $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the Euler products of, essentially, characteristic polynomials of all the Frobenius homomorphisms acting via $\sigma$.

   Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:

   1. for 1-dimensional Galois representations $\sigma$ (hence for $n = 1$) Artin reciprocity produces for each $\sigma$ a Dirichlet character, or more generally a Hecke character $\chi$, and therefrom is built a Dirichlet L-function or Hecke L-function $L_\chi$, respectively, which equals the corresponding Artin L-function $L_\sigma$;

   2. for general $n$-dimensional Galois representations $\sigma$ the conjecture of Langlands correspondence states that there is an automorphic representation $\pi$ corresponding to $\sigma$ and an automorphic L-function $L_\pi$ built from that, which equalso the Artin L-function $L_\sigma$.

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2. • CommentRowNumber2.
   • CommentAuthorAnton Hilado
   • CommentTimeJul 2nd 2022
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   Author: Anton Hilado
   Format: MarkdownItexAdded L-function of a modular form. <a href="https://ncatlab.org/nlab/revision/diff/L-function/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/L-function/18">v18</a>, <a href="https://ncatlab.org/nlab/show/L-function">current</a>

   Added L-function of a modular form.

   diff, v18, current