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

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Abstract

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.

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Notes

  1. As usual \(\varepsilon \) denotes an arbitrarily small positive number.

  2. The cases \(\ell \le 3\) are actually not covered but these cases are superseded by the aforementioned stronger pointwise bounds. And in fact our stronger Theorem 6.2 fully covers the case \(d=2\) of Theorem 1.1 too.

  3. Of course, for \(l=0\) we set \(\mathfrak {p}_1\cdots \mathfrak {p}_l:=\mathscr {O}_F\) and \(\delta _{\mathscr {O}_F}:=1\).

  4. In the cited works the authors used the absolute instead of the relative height, and denoted the invariant by \(\delta (K/k)\) and \(\delta (K)\) respectively.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    Christopher Frei

  2. Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, UK

    Martin Widmer

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  1. Christopher Frei
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  2. Martin Widmer
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Correspondence to Martin Widmer.

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Dedicated to Professor Robert F. Tichy on the occasion of his 60th birthday.

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Frei, C., Widmer, M. Average bounds for the \(\ell \)-torsion in class groups of cyclic extensions. Res. number theory 4, 34 (2018). https://doi.org/10.1007/s40993-018-0127-9

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