The density of discriminants of quartic rings and fields.

*(English)*Zbl 1159.11045In a letter from July 26, 1938, Hasse asked Arnold Scholz what he knew about minimal discriminants of quartic number fields. On July 27, Scholz answered Hasse and mentioned work of J. Mayer [Sitzungsberichte Wien 138, 733–742 (1929; JFM 55.0104.05)] and Delaunay [B. Delaunay, J. Sominski and K. Billevitch, Bull. Acad. Sc. URSS (7) 1935, 1267–1297 (1935; JFM 61.1077.04)]. In the same letter, Scholz also considered the Dirichlet series \(G(s) = \sum D_K^{-s}\), where the sum is over all quartic number fields \(K\) with Galois group \(G\) and \(D_K = | \text{disc}\;K| \). He then claimed that the abscissa of convergence of \(D_4(s)\) (here \(G = D_4\) is the dihedral group of order \(8\)) is \(1\), and those of \(Z(s)\) and \(V(s)\) (counting cyclic and biquadratic extensions) are \(= \frac12\); he remarked moreover that the quotient \(Z(s)/V(s)\) can be extended to the half plane Re \(s > \frac12\), and that \(\lim_{s \to 1/2} Z(s)/V(s) = 0\). Finally he conjectured that \(S_4(s)\) and \(A_4(s)\) have abscissa of convergence equal to \(1\) and \(\frac12\), respectively.

Scholz’s unsecure position, the outbreak of World War II, and his untimely death in 1942 prevented him from publishing any details, and it is not known how Scholz proved his claims. A. M. Baily [J. Reine Angew. Math. 315, 190–210 (1980; Zbl 0421.12007)] later showed that the number of \(D_4\)-extensions with \(D_K < X\) lies between \(c_1X\) and \(c_2X\) for positive constants \(c_1, c_2\) (which is slightly stronger than Scholz’s claim that the abscissa of convergence of \(D_4(s)\) is \(1\)), and H. Cohen, F. Diaz y Diaz and M. Olivier [Compos. Math. 133, No. 1, 65–93 (2002; Zbl 1050.11104)] showed that this number is \(\sim cX\) for an explicitly given constant \(c\). The corresponding questions for abelian extensions were treated by S. Mäki [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 54 (1985; Zbl 0566.12001)], and these results imply the statements of Scholz.

In this article, Bhargava counts the number of \(S_4\)-extensions of \(\mathbb Q\). Let \(N_4^{(j)}(X)\) denote the number of quartic \(S_4\)-extensions with \(4-2j\) real embeddings and \(D_K < X\). Then \(\lim_{X \to \infty} N_4^{(j)}(X)/X = C_j\) exists, and we have \(C_0 = C/48\), \(C_1 = C/8\) and \(C_2 = C/16\) for \(C = \prod_p(1+p^{-2} - p^{-3} - p^{-4})\). These results (and their proofs) also have a number of corollaries. First of all, a positive proportion of quartic fields have Galois group \(D_4\); on the other hand, B. L. van der Waerden [Math. Ann. 109, 13–16 (1933; Zbl 0007.39101; JFM 59.0124.05)] has shown that the density of polynomials with the symmetric group as their Galois group is \(1\) when the polynomials are ordered with respect to the size of their coefficients. Another corollary is the fact that the average size of the \(2\)-class groups of real and complex cubic fields is equal to \(\frac 54\) and \(\frac32\), respectively. These values agree with the original predictions by H. Cohen and J. Martinet [J. Reine Angew. Math. 404, 39–76 (1990; Zbl 0699.12016)], and not with the modifications introduced in [Math. Comput. 63, No. 207, 329–334 (1994; Zbl 0827.11067)] because the original conjectures did not seem to agree with calculations.

The proofs make use of the author’s results on higher composition laws [Ann. Math. (2) 159, No. 1, 217–250 (2004; Zbl 1072.11078)], in particular for pairs of ternary quadratic forms, and the parametrizations of quartic rings [Ann. Math. (2) 159, No. 3, 1329–1360 (2004; Zbl 1169.11045)]. The actual calculations are quite technical; for a survey sketching the ideas behind this approach see K. Belabas [Astérisque 299, 267–299, Exp. No. 935 (2005; Zbl 1090.11066)].

Scholz’s unsecure position, the outbreak of World War II, and his untimely death in 1942 prevented him from publishing any details, and it is not known how Scholz proved his claims. A. M. Baily [J. Reine Angew. Math. 315, 190–210 (1980; Zbl 0421.12007)] later showed that the number of \(D_4\)-extensions with \(D_K < X\) lies between \(c_1X\) and \(c_2X\) for positive constants \(c_1, c_2\) (which is slightly stronger than Scholz’s claim that the abscissa of convergence of \(D_4(s)\) is \(1\)), and H. Cohen, F. Diaz y Diaz and M. Olivier [Compos. Math. 133, No. 1, 65–93 (2002; Zbl 1050.11104)] showed that this number is \(\sim cX\) for an explicitly given constant \(c\). The corresponding questions for abelian extensions were treated by S. Mäki [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 54 (1985; Zbl 0566.12001)], and these results imply the statements of Scholz.

In this article, Bhargava counts the number of \(S_4\)-extensions of \(\mathbb Q\). Let \(N_4^{(j)}(X)\) denote the number of quartic \(S_4\)-extensions with \(4-2j\) real embeddings and \(D_K < X\). Then \(\lim_{X \to \infty} N_4^{(j)}(X)/X = C_j\) exists, and we have \(C_0 = C/48\), \(C_1 = C/8\) and \(C_2 = C/16\) for \(C = \prod_p(1+p^{-2} - p^{-3} - p^{-4})\). These results (and their proofs) also have a number of corollaries. First of all, a positive proportion of quartic fields have Galois group \(D_4\); on the other hand, B. L. van der Waerden [Math. Ann. 109, 13–16 (1933; Zbl 0007.39101; JFM 59.0124.05)] has shown that the density of polynomials with the symmetric group as their Galois group is \(1\) when the polynomials are ordered with respect to the size of their coefficients. Another corollary is the fact that the average size of the \(2\)-class groups of real and complex cubic fields is equal to \(\frac 54\) and \(\frac32\), respectively. These values agree with the original predictions by H. Cohen and J. Martinet [J. Reine Angew. Math. 404, 39–76 (1990; Zbl 0699.12016)], and not with the modifications introduced in [Math. Comput. 63, No. 207, 329–334 (1994; Zbl 0827.11067)] because the original conjectures did not seem to agree with calculations.

The proofs make use of the author’s results on higher composition laws [Ann. Math. (2) 159, No. 1, 217–250 (2004; Zbl 1072.11078)], in particular for pairs of ternary quadratic forms, and the parametrizations of quartic rings [Ann. Math. (2) 159, No. 3, 1329–1360 (2004; Zbl 1169.11045)]. The actual calculations are quite technical; for a survey sketching the ideas behind this approach see K. Belabas [Astérisque 299, 267–299, Exp. No. 935 (2005; Zbl 1090.11066)].

Reviewer: Franz Lemmermeyer (Jagstzell)

##### MSC:

11R45 | Density theorems |

11R16 | Cubic and quartic extensions |

11R29 | Class numbers, class groups, discriminants |