부산대학교 도서관

Let kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk be a number field and kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk be a finite group. Let kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk be the family of number fields kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk with absolute discriminant kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk at most kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk such that kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk is normal with Galois group isomorphic to kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk. If kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk is the symmetric group kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk or any transitive group of prime degree, then we unconditionally prove that for all kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk with at most kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk exceptions, the kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk-functions associated to the faithful Artin representations of kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnkkGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;for a prime kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk, the periodic torus orbits attached to the ideal classes of almost all totally real degree kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk fields kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk over ℚ equidistribute on kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk with respect to Haar measure;for each kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk, the kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk-torsion subgroups of the ideal class groups of almost all degree kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk fields over kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk (resp. almost all degree kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnkkGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk-fields over kGFkG(Q)KDKQK/kGGSnK∈FkG(Q)Oε(Qε)LGal(K/k)nSnppFPGLp(Z)∖PGLp(R)ℓ≥2ℓpknSnk) are as small as GRH implies; andan effective variant of the Chebotarev density theorem holds for almost all fields in such families.