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Artin L-Functions Artin L-functions Charlotte Euvrard Laboratoire de Mathématiques de Besançon January 10, 2014 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 1 / 12 Definition L=K Galois extension of number fields G = Gal(L=K ) its Galois group Let (r;V ) be a representation of G and c the associated character. Let p be a prime ideal in the ring of integers OK of K , we denote by : Ip the inertia group jp the Frobenius automorphism I V p = fv 2 V : 8i 2 Ip; r(i)(v) = vg Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 2 / 12 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Sum and Euler product Proposition Let d and d0 be the degrees of the representations (r;V ) and (r~;V Ip ). Let ai;r(p), 1 6 i 6 d be the eigenvalues of r~(jp) with ai;r(p) = 0 if 0 d < i 6 d. Then : d −s−1 −s L(s;c;L=K ) = ∏ ∏ 1 − ai;r(p)p = ∑ ac(n)n : p2Z i=1 n>1 0 In particular, jai;r(p)j = 1 for 1 6 i 6 d . Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 4 / 12 Properties Properties 1 We have : L(s;1G;L=Q) = z(s); where z is the Riemann zeta function. 2 For c1, c2 two characters of G, L(s;c1 + c2;L=Q) = L(s;c1;L=Q)L(s;c2;L=Q): 3 We can write : c(1) zL(s) = z(s) ∏ L(s;c;L=Q) c6=1 where zL is the Dedekind zeta function of L. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 5 / 12 Examples Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 6 / 12 Functional equation Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 7 / 12 Gamma factors Definition Define : − dimV + dimV − s s − s+1 s + 1 gc(s) = p 2 Γ p 2 Γ 2 2 for every w of L above the infinite place of Q : the generator sw of G(w) = fg 2 G j r(g)(w) = wg induces an eigenspace decomposition V = V + ⊕ V − where V + and V − correspond to the eigenvalues +1 and −1 of r(sw ). Actually, + <1;sw > 1 dimV = dimV = 2 (d + c(sw )) and so − 1 dimV = 2 (d − c(sw )). Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 8 / 12 Artin conductor Definition Let Gi be the i-th ramification group of p above p 2 Z. Define : +¥ jG j i Gi fp(c) = ∑ codimV : i=0 jG0j Definition (Artin conductor) The Artin conductor is : f (c) = ∏ pfp(c): p2Z Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 9 / 12 Property We can write : − dimV + dimV − ds s s + 1 gc(s) = p 2 Γ Γ : 2 2 Functional equation Theorem s Functional equation Let Λ(s;c) = f (c) 2 gc(s)L(s;c;L=Q) be the completed Artin L-function. Then Λ(s;c) admits a meromorphic continuation to C, analytic except for poles at s = 0 and s = 1 and satisfies : Λ(1 − s;c) = W (c)Λ(s;c); with a constant W (c) of absolute value 1. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 10 / 12 Functional equation Theorem s Functional equation Let Λ(s;c) = f (c) 2 gc(s)L(s;c;L=Q) be the completed Artin L-function. Then Λ(s;c) admits a meromorphic continuation to C, analytic except for poles at s = 0 and s = 1 and satisfies : Λ(1 − s;c) = W (c)Λ(s;c); with a constant W (c) of absolute value 1. Property We can write : − dimV + dimV − ds s s + 1 gc(s) = p 2 Γ Γ : 2 2 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 10 / 12 The nontrivial zeroes of L(s;c) lie in the open strip fs 2 C : 0 < Re(s) < 1g. 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;c) is entire. Conjectures The trivial zeroes of L(s;c) occur at negative integers. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12 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;c) is entire. Conjectures The trivial zeroes of L(s;c) occur at negative integers. The nontrivial zeroes of L(s;c) lie in the open strip fs 2 C : 0 < Re(s) < 1g. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12 Artin conjecture For each nontrivial irreducible character, L(s;c) is entire. Conjectures The trivial zeroes of L(s;c) occur at negative integers. The nontrivial zeroes of L(s;c) lie in the open strip fs 2 C : 0 < Re(s) < 1g. 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 Conjectures The trivial zeroes of L(s;c) occur at negative integers. The nontrivial zeroes of L(s;c) lie in the open strip fs 2 C : 0 < Re(s) < 1g. 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;c) is entire. Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 11 / 12 Examples Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 12 / 12.
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