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arXiv · 2506.05539

The second moment of the size of the $2$-class group of monogenized cubic fields

Manjul Bhargava·Arul Shankar·Ashvin Swaminathan
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Abstract

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.

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Manjul Bhargava, Arul Shankar, Ashvin Swaminathan. 2025-06-05. The second moment of the size of the $2$-class group of monogenized cubic fields. https://arxiv.org/abs/2506.05539

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