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The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields
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abstract
We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.
Forward citations
Cited by 4 Pith papers
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Hecke reciprocity and class groups
The paper proves the average size of 2-torsion in class groups is 2 for one 3-adic ramification family of pure cubic fields and 3/2 for the other, and attributes the split to Hecke reciprocity.
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Secondary terms in the first moment of $|{\rm Sel}_2(E)|$
For elliptic curves ordered by height, the first moment of |Sel2(E)| contains a secondary term of order X^{3/4}, with a power-saving error of size O(X^{3/4-1/3804+epsilon}).
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A parametrization of $3$-class groups of quadratic rings over Dedekind domains
Orbits of binary cubic forms under GL(R ⊕ a) are canonically in bijection with 3-torsion ideal classes of quadratic rings over any Dedekind domain R with char(R) ≠ 3, with the Steinitz class [a] as an explicit parameter.
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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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