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

Christopher Frei*, Martin Widmer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)1205-1229
Number of pages25
JournalMathematische Annalen
Issue number3-4
Early online date23 Dec 2020
Publication statusPublished - 1 Apr 2021

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

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