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Averages and higher moments for the $\ell$-torsion in class groups. (arXiv:1810.04732v1 [math.NT])
来源于:arXiv
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 bou 查看全文>>