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arXiv · 1606.06103

On $\ell$-torsion in class groups of number fields

Jordan Ellenberg·Lillian B. Pierce·Melanie Matchett Wood
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Abstract

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.

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Jordan Ellenberg, Lillian B. Pierce, Melanie Matchett Wood. 2016-06-20. On $\ell$-torsion in class groups of number fields. https://doi.org/10.2140/ant.2017.11.1739

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