On class numbers of pure quartic fields

Li, Jianing; Xu, Yan

doi:10.1007/s11139-020-00253-2

Cited by 2 publications

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“…The following proposition collects results about the 2-class groups Cl 2 (−l) and Cl 2 (−2l) due to Gauss, Hasse [10], Brown [2] and others, most importantly due to Leonard-Williams [15], see [16,Theorem 4.2] for a proof about Cl 2 (−2l). Proposition 3.6.…”

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confidence: 96%

On abelian $2$-ramification torsion modules of quadratic fields

Li¹,

Ouyang²,

Xu³

2020

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For a number field F and a prime number p, the Zp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F , denoted as Tp(F ), is an important subject in abelian p-ramification theory. In this paper we study the groupFirstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk 4 (T 2 (−m)) = rk 2 (T 2 (−m))−rank(R) where R is a certain explicitly described Rédei matrix over F 2 . Furthermore, applying this formula and exploring the connection of T 2 (−m) to the ideal class group of Q( √ −m) and the tame kernel of Q( √ m), we obtain the 4-rank density of T 2 -groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-divisibility of orders of T 2 (±l) and T 2 (±2l), both of which are cyclic 2-groups. In particular we find that #T 2 (l)We then obtain density results for T 2 (±l) and T 2 (±2l) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about Tp(F ) when F varies over real or imaginary quadratic fields for any prime p, and about T 2 (±l) and T 2 (±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T 2 (l) case is closely connected to Shanks-Sime-Washington's speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.

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confidence: 96%