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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2014
    • (edited Aug 28th 2014)

    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 σ:GalGL n()\sigma \colon Gal \longrightarrow GL_n(\mathbb{C}) as the Euler products of, essentially, characteristic polynomials of all the Frobenius homomorphisms acting via σ\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 σ\sigma (hence for n=1n = 1) Artin reciprocity produces for each σ\sigma a Dirichlet character, or more generally a Hecke character χ\chi, and therefrom is built a Dirichlet L-function or Hecke L-function L χL_\chi, respectively, which equals the corresponding Artin L-function L σL_\sigma;

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

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