论文标题
Artin的Holomorphy猜想和Artin $ L $ functions的一种大致形式
An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions
论文作者
论文摘要
令$ k $为一个数字字段，$ g $为有限组。令$ \ mathfrak {f} _ {k}^{g}（q）$是数字字段$ k $的家族$ k $，最多$ k $，$ k/k $，以便$ k/k $与galois group isomorphic isomorphic to $ g $正常。如果$ g $是对称群体$ s_n $或任何跨性别的质量级别，那么我们无条件地证明，对于所有$ k \ in \ mathfrak {f} _k^g（q）$，最多与$ o_is（q^ε）$异常，$ l $ unctions $ o_is（q^ε），$ l $ - $ l $ a $ a $ a $ a $ a $ n Math Art a gallm Mathave（K）（K）（KAL）（KAL）（KAL）（KAL）（k）与Artin猜想和普遍的Riemann假设相称的全体形状和非逐渐变化。该结果是更一般定理的特殊情况。作为应用程序，我们证明： 1）在$ \ mathbb {q} $上，其类组与Artin猜想和GRH所暗示的，在$ \ mathbb {q} $上面存在无限的多大度$ n $ n $ n $ s_n $ - fields。 2）对于Prime $ p $，几乎全部真正的$ P $ f $ field $ \ mathbb {q} $ equidistribute上的理想类别附加的定期圆环轨道，上关于HAAR措施； 3）对于每个$ \ ell \ geq 2 $，几乎所有学位的$ p $ p $ p $ p $ p $ k $的$ \ ell $ torsion子组（分别是$ k $的$ p $ fields）（分别几乎所有$ n $ n $ n $ n $ s_n $ fields aver $ k $上的$ k $）都与grh inmans一样小;和 4）Chebotarev密度定理的有效变体适用于此类家庭的几乎所有领域。
Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the symmetric group $S_n$ or any transitive group of prime degree, then we unconditionally prove that for all $K\in\mathfrak{F}_k^G(Q)$ with at most $O_ε(Q^ε)$ exceptions, the $L$-functions associated to the faithful Artin representations of $\mathrm{Gal}(K/k)$ 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: 1) there exist infinitely many degree $n$ $S_n$-fields over $\mathbb{Q}$ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; 2) for a prime $p$, the periodic torus orbits attached to the ideal classes of almost all totally real degree $p$ fields $F$ over $\mathbb{Q}$ equidistribute on $\mathrm{PGL}_p(\mathbb{Z})\backslash\mathrm{PGL}_p(\mathbb{R})$ with respect to Haar measure; 3) for each $\ell\geq 2$, the $\ell$-torsion subgroups of the ideal class groups of almost all degree $p$ fields over $k$ (resp. almost all degree $n$ $S_n$-fields over $k$) are as small as GRH implies; and 4) an effective variant of the Chebotarev density theorem holds for almost all fields in such families.
