Research in Number Theory

, 4:34 | Cite as

Average bounds for the \(\ell \)-torsion in class groups of cyclic extensions

  • Christopher Frei
  • Martin WidmerEmail author


For all positive integers \(\ell \), we prove non-trivial bounds for the \(\ell \)-torsion in the class group of K, which hold for almost all number fields K in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-p-extensions of F, where F is an arbitrary number field and p is any prime for which F and the pth cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.


\(\ell \)-torsion Class group Number fields Small height 

Mathematics Subject Classification

Primary 11R29 11N36 11R45 Secondary 11G50 


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© SpringerNature 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK
  2. 2.Department of Mathematics, Royal HollowayUniversity of LondonEghamUK

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