Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are significantly better than previously known asymptotic bounds. Applied to the integers, they express that any subgroup of index $i$ of the multiplicative group of integers modulo $m$ is generated by prime numbers smaller than $16(i\log m)^2$, subject to the Riemann Hypothesis. Two particular consequences relate to mathematical cryptology. Applied to cyclotomic fields, they provide explicit bounds on generators of the relative class group, needed in some previous work on the shortest vector problem on ideal lattices. Applied to Jacobians of hyperelliptic curves, they allow one to derive bounds on the degrees of isogenies required to make their horizontal isogeny graphs connected. Such isogeny graphs are used to study the discrete logarithm problem on said Jacobians.
设$K_{m}=\Bbb{Q}(ζ_{m})$，其中$ζ_{m}$是本原$m$的单位根.令$p>2$是素数，令$C_{p}$表示阶为$p.$的群$K_{m}$的代数整数环是$\Cal{O}_{m}=\Bbb{Z}[ζ_{m}].$令$Λ_{m,p}$表示代数$K_{m}[C_{p}].$中的阶$\Cal{O}_{m}[C_{p}]$考虑核群$D(Λ_{m,p})$和Swan子群$T(Λ_{m,p}).$如果$(p,m)=1$这两个子群的类群重合。限制到当有一个有理素数$p$在$\Cal{O}_{m}$中是素数时，需要$m=4$或$q^{n}$，其中$q>2$是素数。对于每一个这样的$m$，$3 \leq m \leq 100,$，我们给这样一个素数，并表明，人们可以计算$T(Λ_{m,p})$作为一个有限域的单位群的商。当$h_{mp}^{+}=1$时，我们给出了$|T(Λ_{m,p})|$的精确值，对于其他情况，我们提供了一个上限。我们探讨这些结果的伽罗瓦模理论的影响。
The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias
School of Mathematics, Nanjing University（南京大学数学系）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心；中国科学院大学）；University of the Chinese Academy of Sciences
