Artin L-functions. Charlotte Euvrard. January 10, Laboratoire de Mathématiques de Besançon

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1 Artin L-functions Charlotte Euvrard Laboratoire de Mathématiques de Besançon January 10, 2014 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 1 / 12

2 Definition L/K Galois extension of number fields G = Gal(L/K ) its Galois group Let (ρ,v) be a representation of G and χ the associated character. Let p be a prime ideal in the ring of integers O K of K, we denote by : I p the inertia group ϕ p the Frobenius automorphism V I p = {v V : i I p, ρ(i)(v) = v} Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 2 / 12

3 Definition Definition The Artin L-function is defined by : 1 L(s,χ,L/K ) =, Re(s) > 1, p O K det(id N(p) s ϕ p ;V I p) where det(id N(p) s ρ(ϕ p )) if p is unramified det(id N(p) s ϕ p ;V I p ) = det(id N(p) s ρ(ϕ p )) if p is ramified and V Ip {0} 1 otherwise and ρ( σ) = 1 I p j I p ρ(jσ) Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12

4 Definition Definition The Artin L-function is defined by : 1 L(s,χ,L/K ) =, Re(s) > 1, p O K det(id N(p) s ϕ p ;V I p) where det(id N(p) s ρ(ϕ p )) if p is unramified det(id N(p) s ϕ p ;V I p ) = det(id N(p) s ρ(ϕ p )) if p is ramified and V Ip {0} 1 otherwise and ρ( σ) = 1 I p j I p ρ(jσ) 1 In the particular case K = Q : L(s,χ,L/Q) = det(id p s ϕ p ;V I p ) p Z Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12

5 Sum and Euler product Proposition Let d and d be the degrees of the representations (ρ,v) and ( ρ,v I p ). Let α i,ρ (p), 1 i d be the eigenvalues of ρ(ϕ p ) with α i,ρ (p) = 0 if d < i d. Then : L(s,χ,L/K ) = d p Z i=1 In particular, α i,ρ (p) = 1 for 1 i d. ( 1 αi,ρ (p)p s) 1 = a χ (n)n s. n 1 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 4 / 12

6 Properties Properties 1 We have : L(s,1 G,L/Q) = ζ(s), where ζ is the Riemann zeta function. 2 For χ 1, χ 2 two characters of G, 3 We can write : L(s,χ 1 + χ 2,L/Q) = L(s,χ 1,L/Q)L(s,χ 2,L/Q). ζ L (s) = ζ(s) L(s,χ,L/Q) χ(1) χ 1 where ζ L is the Dedekind zeta function of L. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 5 / 12

7 Examples Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 6 / 12

8 Functional equation Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 7 / 12

9 Gamma factors Definition Define : γ χ (s) = ( ( π s s )) ( dimv + ( 2 Γ π s+1 s Γ 2 2 )) dimv for every w of L above the infinite place of Q : the generator σ w of G(w) = {g G ρ(g)(w) = w} induces an eigenspace decomposition V = V + V where V + and V correspond to the eigenvalues +1 and 1 of ρ(σ w ). Actually, dimv + = dimv <1,σ w> = 1 2 (d + χ(σ w)) and so dimv = 1 2 (d χ(σ w)). Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 8 / 12

10 Artin conductor Definition Let G i be the i-th ramification group of p above p Z. Define : + G i f p (χ) = i=0 G 0 codimv G i. Definition (Artin conductor) The Artin conductor is : f(χ) = p fp(χ). p Z Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 9 / 12

11 Functional equation Theorem Functional equation Let Λ(s,χ) = f(χ) s 2 γ χ (s)l(s,χ,l/q) be the completed Artin L-function. Then Λ(s, χ) admits a meromorphic continuation to C, analytic except for poles at s = 0 and s = 1 and satisfies : Λ(1 s,χ) = W(χ)Λ(s,χ), with a constant W(χ) of absolute value 1. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 10 / 12

12 Functional equation Theorem Functional equation Let Λ(s,χ) = f(χ) s 2 γ χ (s)l(s,χ,l/q) be the completed Artin L-function. Then Λ(s, χ) admits a meromorphic continuation to C, analytic except for poles at s = 0 and s = 1 and satisfies : Λ(1 s,χ) = W(χ)Λ(s,χ), with a constant W(χ) of absolute value 1. Property We can write : ( γ χ (s) = π ds s ) dimv + ( s Γ Γ 2 2 ) dimv. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 10 / 12

13 Conjectures The trivial zeroes of L(s, χ) occur at negative integers. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12

14 Conjectures The trivial zeroes of L(s, χ) occur at negative integers. The nontrivial zeroes of L(s,χ) lie in the open strip {s C : 0 < Re(s) < 1}. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12

15 Conjectures The trivial zeroes of L(s, χ) occur at negative integers. The nontrivial zeroes of L(s,χ) lie in the open strip {s C : 0 < Re(s) < 1}. Grand Riemann Hypothesis The nontrivial zeros of Artin L-functions lie on the critical line 1/2 + it. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12

16 Conjectures The trivial zeroes of L(s, χ) occur at negative integers. The nontrivial zeroes of L(s,χ) lie in the open strip {s C : 0 < Re(s) < 1}. Grand Riemann Hypothesis The nontrivial zeros of Artin L-functions lie on the critical line 1/2 + it. Artin conjecture For each nontrivial irreducible character, L(s, χ) is entire. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12

17 Examples Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 12 / 12