On ℓ-torsion in class groups of number fields

By Jordan S. Ellenberg, L. Pierce, Melanie Matchett Wood. 2016

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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.

Subfields: Algebra and Number Theory, Artificial Intelligence, Information Systems

View Paper: https://www.semanticscholar.org/paper/ba43281d718c9d5e39a041b48bc760589b5f2149