Averages and higher moments for the ℓ -torsion in class groups

Christopher Frei*, Martin Widmer

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

Abstract

We prove upper bounds for the average size of the ℓ-torsion ClK[ℓ] of the class group of K, as K runs through certain natural families of number fields and ℓ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for ClK[ℓ] 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 ClK[ℓ] for many families of number fields K considered in the literature, for example, for the families of all degree-d-fields for d∈{2,3,4,5} (and non-D4 if d=4). As an application of the case d=2 we obtain the best upper bounds for the number of Dp-fields of bounded discriminant, for primes p>3.
Originalspracheenglisch
Seiten (von - bis)1205-1229
Seitenumfang25
FachzeitschriftMathematische Annalen
Jahrgang379
Ausgabenummer3-4
Frühes Online-Datum23 Dez 2020
DOIs
PublikationsstatusVeröffentlicht - 1 Apr 2021

ASJC Scopus subject areas

  • !!Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

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