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• W3100503615 abstract "Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $textrm{Gal}(L/mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $textrm{Cl}^{+}(K)[2]$." @default.
• W3100503615 created "2020-11-23" @default.
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• W3100503615 date "2020-08-24" @default.
• W3100503615 modified "2023-10-16" @default.
• W3100503615 title "Higher genus theory" @default.
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• W3100503615 doi "https://doi.org/10.1093/imrn/rnaa196" @default.
• W3100503615 hasPublicationYear "2020" @default.
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