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Monogenicity and 2-torsion in the class group of number fields of odd degree
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abstract
We study the average $2$-torsion in the class group of monogenised fields of odd degree. Bhargava--Hanke--Shankar have recently shown that for a fixed signature, the average number of non-trivial $2$-torsion elements in the class group of monogenised cubic fields is exactly twice the value predicted by the Cohen--Lenstra--Martinet--Malle heuristic over the full $S_3$ family. For any odd degree $n \ge 3$ and signature, we prove that the average number of non-trivial $2$-torsion elements in the class group of monogenised fields is at most twice the value predicted by the Cohen--Lenstra--Martinet--Malle heuristic over the full $S_n$ family. Conditional on a tail estimate for $n \ge 5$, this establishes that the doubling phenomenon discovered by Bhargava--Hanke--Shankar persists across all odd degrees and signatures.
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The second moment of the size of the $2$-class group of monogenized cubic fields
The second moment of the size of the 2-class group of monogenized cubic fields is at most 3 for totally real fields and at most 6 for complex fields, with matching lower bounds conditional on a tail estimate.
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