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

2020/10/13 by Jiuya Wang · 1 citation

paper · doi:10.1515/crelle-2020-0034

crossref issued 2020/10/13 · crossref published 2020/10/13 · crossref published-online 2020/10/13 · crossref created 2020/10/14 · crossref published-print 2021/04/01 · crossref deposited 2026/04/17 · crossref indexed 2026/07/30

Abstract

Abstract Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r A=(ℤ/pℤ)^r for a prime number p . For every integer ℓ > 1 ℓ>1 and r > 1 r>1 , we prove a non-trivial upper bound on the ℓ ℓ -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 ℓ ℓ -torsion in class groups are bounded non-trivially for every G -extension and every integer ℓ > 1 ℓ>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.

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