Chapter24Artin \(L\)-functions and the Chebotaryov density theorem
This unit begins the fourth and final part of the course. In this part, we describe some other types of \(L\)-functions that are used for arithmetic purposes. This merely scratches the surface of what is now a rather vast theory of \(L\)-functions; \S 5 of [9] gives a somewhat less narrow account. Some of this discussion will only make sense if you have studied some algebraic number theory. The book I used to teach 18.786 last year is a reasonable place to start: it is Janusz, \textit{Algebraic Number Fields}. I'm also presuming you are happy with representation theory of finite groups at the level of 18.702.\(K\)\(\QQ\text{,}\)\(G = \Gal(K/\QQ)\text{.}\)\(\gotho_K\)\(K\text{;}\)\(\alpha \in \gotho_K\)\(\alpha\)\(\ZZ\text{.}\)\(p\)\(K\)\(\gotho_K/p \gotho_K\)\(K = \QQ(i)\text{,}\)\(\gotho_K = \ZZ[i]\text{,}\)\(p=2\text{.}\)\(K/\QQ\text{.}\)\(\gotho_K/p \gotho_K\text{,}\)\(G\)\(x \mapsto x^p\text{.}\)\(g \in G\)\(x^p = x^g\)\(x \in \gotho_K/p \gotho_K\text{.}\)\(p\)\(g \in G\)\(g\)\(p\text{.}\)\(K\)\(\CC\text{,}\)\(\CC\)\(K\text{.}\)\(K\text{.}\)\(L\)\(\rho: G \to \GL_n(\CC)\)\(\chi: G \to \CC\text{;}\)\(\chi(g) = \Trace \rho(g)\text{.}\)\(L\)\(\rho\)\(p\)\(\Frob_p\)\(p\text{;}\)\(\rho\)\(p\text{.}\)\(\rho: G \to \GL_1(\CC)\)\(L(\rho,s)\)\(K'\)\(\QQ\)\(K\text{,}\)\(\rho\)\(\Gal(K'/\QQ)\text{,}\)\(L\)\(K\)\(K'\)\(p\)\(\rho\text{.}\)\(\Real(s) > 1\text{,}\)\(\Real(s) \geq 1 + \epsilon\text{,}\)\(L(\rho,s)\)\(\CC\text{,}\)\(s=1\text{,}\)\(s=1\)\(\rho\)\(1/|G| \sum_{g \in G} \chi(g)\)\(L(\rho,s)\)\(L(\overline{\rho}, 1-s)\text{.}\)\(L(\rho,s)\)\(\theta\)\(\rho\)\(\rho\)\(\rho\)\(L\)\(\rho\)\(\zeta\)\(\rho\)\(\rho\)\(\rho\)\(A_5\text{.}\)\(\rho\)\(-1\)\(H\)\(G\)\(\sigma: H \to \GL_m(\CC)\)\(V\)\(f: G \to \CC^m\)\(h(f(g)) = f(hg)\)\(h \in H\text{.}\)\(G\)\(\Ind^G_H(\sigma)\text{.}\)\(\sigma\)\(\Ind^G_H(\sigma)\)\(G\)\(H\text{.}\)\(G\)\(\QQ\)\(G\text{,}\)\(g\)\(H\text{.}\)\(H\)\(g\text{;}\)\(G\)\(\rho\text{,}\)\(L\)\(L(\rho,s)\)\(\Real(s) \geq 1\text{.}\)\(\Real(s) = 1\text{,}\)\(L(\rho,s)\)\(s \neq 1\text{,}\)\(s=1\text{,}\)\(L(\rho,s)\)\(- 1/|G| \sum_{g \in G} \chi(g)\text{.}\)\(s=1\)\(\rho\text{.}\)\(\rho = \Ind^G_H \sigma\)\(H\)\(G\)\(\sigma: H \to \GL_1(\CC)\text{.}\)\(\sigma\)\(\rho\text{,}\)\(\rho^{\oplus m}\text{.}\)\(\rho\text{,}\)\(\rho\)\(L(\rho,s)^m\)\(\Real(s) \geq 1\text{,}\)\(m\)\(s\)\(\Real(s) = 1\text{;}\)\(\Real(s) > 1\text{.}\)\(C\)\(G\text{,}\)\(p\)\(\Frob_p \in C\)\(\#C/\#G\text{.}\)\(G\)\(S_n\)\(G\)\(\QQ\)\(G\text{,}\)\(P\)\(G\text{.}\)\(P\)\(d_1, \dots, d_k\text{.}\)
