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Comparing the density of \(D_4\) and \(S_4\) quartic extensions of number fields. (English) Zbl 1467.11101

Let \(F\) be an algebraic number field \(F\) of degree \(d\), discriminant \(D_F\) and signature \((r_1,r_2)\). For a finite group \(G\) let \(N_n^F(x;G)\) be the number of extensions \(K/F\) of degree \(n\) whose Galois closure has \(G\) for its Galois group.
For the dihedral group \(D_4\) H. Cohen et al. [Compos. Math. 133, No. 1, 65–93 (2002; Zbl 1050.11104)] established \[N_4^F(x;D_4) = (a_F+o(1))x\tag{D4}\] with \[ a_F=\sum_L\frac{R_L}{2^{r_2(L)+1}D(L/F)^2\zeta_L(2)}, \] where \(L\) runs over all quadratic extensions of \(F\), \(R_L\) is the residue at \(s=1\) of \(\zeta_L(s)\) and \(D(L/F)\) is the norm of the relative discriminant of \(L/F\).
For the symmetric group \(S_4\) one has
\[N_4^F(x;S_4) = (b_F+o(1))x\tag{S4}\] with \[ b_F = \frac{10^{r_1}R_F}{2\cdot 24^{r_1+r_2}}\prod_\mathfrak p\left(1+\frac1{N\mathfrak p^2}-\frac1{N\mathfrak p^3}-\frac1{N\mathfrak p^4}\right) \] [M. Bhargava et al., “Geometry-of-numbers methods over global fields. I: Prehomogeneous vector spaces”, Preprint, arXiv:1512.03035].
The authors study the ratio \[ A_F(x)=\frac{N_4^F(x;D_4)}{N_4^F(x;S_4)} \] and use (D4) and (S4) to prove that \(GRH\) implies the bound \[A_F(x)\gg_d\frac{g(F)-1}{(\log\log(|D_F|))^d},\tag{ratio}\] where \(g(F)\) is the number of elements of order dividing \(2\) in the class-group of \(F\) (Theorem 1.3) (note that in the formulation of Theorem 1.3 in the paper the summand “\(-1\)” in the numerator has been omitted).
Unconditionally they prove (Theorem 1.1) that for almost all quadratic fields \(F\) and any \(\varepsilon>0\) one has \[ A_F(x)\gg_\varepsilon\log^{2-\varepsilon}(|D_F|). \]

MSC:

11R16 Cubic and quartic extensions
11R42 Zeta functions and \(L\)-functions of number fields
11R29 Class numbers, class groups, discriminants
11R45 Density theorems

Citations:

Zbl 1050.11104
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References:

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