Effect of monogenicity on 2-torsion in the class group of number fields of odd degree

2020/11/17 by Siad, Artane · 1 citation
#11E76 #11R29 #11R45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2011.08834

Abstract

We study average 2-torsion in the class group of monogenised fields of odd degree. Bhargava-Hanke-Shankar have recently shown that the average number of non-trivial 2-torsion elements in the class group of monogenised cubic fields of a fixed signature is twice the value predicted by the Cohen-Lenstra-Martinet-Malle heuristic for the full family. Fix any odd degree n and signature (r₁,r₂). In this work, we prove that the average number of non-trivial elements in the class group of monogenised fields of degree n and signature (r₁,r₂) is bounded by twice the value predicted by Cohen-Lenstra-Martinet-Malle. Conditional on a tail estimate for n ≥ 5, this shows that the doubling phenomenon uncovered for cubic fields by Bhargava-Hanke-Shankar persists across all odd degrees and signatures.

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• Hecke reciprocity and class groups

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