Artin L Functions

Ehud de Shalit

Chapter 4 in An Introduction to the Langlands Program, 2004, pp 73-87 from Springer

Abstract: Abstract Let χ : (ℤ/mℤ)× → ℂ× be a primitive Dirichlet character modulo m. Let K = ℚ(ζ), where ζ = e 2πi/m . The identification G = Gal(K/ℚ) ≃ (ℤ/mℤ)× allows us to attach to χ a character χ Gal : G → ℂ× satisfying 1.1 if (p, m) = 1 and σ p is the Frobenius automorphism at p (the canonical generator of the decomposition group of p in G, which induces on the residue field of any prime of K above p the automorphism x ↦ x p .) The Kronecker-Weber theorem (Kronecker 1853, Weber 1886) asserts that every 1-dimensional character of G ℚ = Gal(̄ℚ/ℚ) is of the form χ Gal for an appropriate χ.

Keywords: Modular Form; Riemann Hypothesis; Class Field Theory; Meromorphic Continuation; Archimedean Place (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-8176-8226-2_4

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