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The number of quartic \(D_4\)-fields with monogenic cubic resolvent ordered by conductor. (English) Zbl 1466.11080

M. M. Wood [Int. Math. Res. Not. 2012, No. 6, 1300–1320 (2012; Zbl 1254.11094)] presented a parametrization of quartic rings with monogenic cubic resolvents in terms of binary quartic forms. The authors use this classification to describe asymptotics of \(N_{D_4}^{r_2}(x)\), the number of isomorphy classes of quartic fields with Galois group \(D_4\) of their Galois closure, having monogenic cubic cubic resolvents, \(2r_2\) complex embeddings and \(C(L)\le x\), where \[ C(L) = \mathrm{disc}(L)/\mathrm{disc}(K_L) \] (with \(K_L\) being the unique quadratic subfield of \(L\)) is the conductor of \(L\). The main result (Theorem 1.3) asserts \[ N_{D_4}^{r_2}(x) = c(r_2)\frac{\sqrt2}{\sqrt\pi}\frac{\Gamma(1/4)^2}{48\zeta(2)}\prod_p\left(1-\frac{2p-1}{p^2}\right)x^{3/4}\log x + O\left((x\log x)^{3/4}\right), \] where \[ c(r_2) = \begin{cases}1&\mathrm{ if\ } r_2=0\\ \sqrt2&\mathrm{ if\ } r_2=1\\ 1+\sqrt2&\mathrm{ if\ } r_2=2.\end{cases} \] In the case when the quartic fields are ordered according to the discriminant the authors present only lower and upper bounds (Theorem 1.7), but if one replaces the group \(D_4\) by the Klein group \(V_4\), then in Theorem 1.3 the asymptotics \[ \frac78\frac{\Gamma(1/3)^2}{\Gamma(2/3)}\prod_p\left(1-\frac{4p-3}{p^3}\right)x^{1/3}+O\left(x^{1/3}(\log x\log\log x)^{-1}\right) \] is obtained.
In the last result (Theorem 1.9) it is shown that the number of elliptic curves \[ E_{a,b}:\ y^2=x(x^2+ax+b) \] (where \(a,b\in Z\) and for any prime \(p\) \(p^2\mid a\) implies \(p^4\nmid b\)) with \(|\mathrm{disc}(E_{a,b})|<x\) equals \[ \frac2{3\zeta(6)}x^{1/2}\log x+O\left(x^{1/2}\right). \]

MSC:

11R16 Cubic and quartic extensions
11E76 Forms of degree higher than two
11G05 Elliptic curves over global fields
11N45 Asymptotic results on counting functions for algebraic and topological structures
11R04 Algebraic numbers; rings of algebraic integers
11R45 Density theorems

Citations:

Zbl 1254.11094

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References:

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