We prove upper bounds for the average size of the $\ell$-torsion $\Cl_K[\ell]$ of the class group of $K$, as $K$ runs through certain natural families of number fields and $\ell$ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for $\Cl_K[\ell]$ to the existence of many primes splitting completely in $K$ that are small compared to the discriminant of $K$. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of $\Cl_K[\ell]$ for many families of number fields $K$ considered in the literature, for example, for the families of all degree-$d$-fields for $d\in\{2,3,4,5\}$ (and non-$D_4$ if $d=4$). As an application of the case $d=2$ we obtain the best upper bounds for the number of $D_p$-fields of bounded discriminant, for primes $p>3$.

中文翻译：

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班级组中 $$\ell $$-torsion 的平均值和更高时刻

我们证明了 $K$ 的类群的 $\ell$-torsion $\Cl_K[\ell]$ 的平均大小的上限，因为 $K$ 穿过某些自然的数域族和 $\ell$是一个正整数。我们改进了一个关键参数，几乎用在所有这种类型的结果中，它将 $\Cl_K[\ell]$ 的上限与在 $K$ 中完全分裂的许多质数的存在联系起来，这些质数与 $K 的判别式相比很小$. 我们的改进是通过引入一个新的数字字段专用不变量系列来替代该论证中的判别式，并结合这些不变量的新计数结果来实现的。这导致文献中考虑的许多数字字段 $K$ 的平均和有时甚至更高的 $\Cl_K[\ell]$ 矩的上限显着提高，例如，对于 $d\in\{2,3,4,5\}$ 的所有 degree-$d$-fields 的家庭（如果 $d=4$，则为非 $D_4$）。作为 $d=2$ 情况的应用，对于素数 $p>3$，我们获得了有界判别式的 $D_p$-域数量的最佳上限。