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Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

  • Jiuya Wang ORCID logo

Abstract

Elementary abelian groups are finite groups in the form of A = ( / p ) r for a prime number p. For every integer > 1 and r > 1 , we prove a non-trivial upper bound on the -torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the -torsion in class groups are bounded non-trivially for every G-extension and every integer > 1 . When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Funding statement: The author is supported by Foerster–Bernstein Fellowship at Duke University.

Acknowledgements

I would like to thank Jürgen Klüners, Weitong Wang and Asif Zaman for providing helpful references. I would like to thank Dimitris Koukoulopoulos, Robert J. Lemke Oliver, Melanie Matchett Wood, Asif Zaman and Ruixiang Zhang for helpful conversations. I would like to thank Jordan Ellenberg, Melanie Matchett Wood and Yongqiang Zhao for suggestions on an earlier draft.

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Received: 2020-03-13
Revised: 2020-09-07
Published Online: 2020-10-13
Published in Print: 2021-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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