Densities of quartic fields with even Galois groups

Abstract

Abstract. Let N (d, G, X) be the number of degree d number fields K with Galois group G and whose discriminant D K satisfies |D K | ≤ X. Under standard conjectures in diophantine geometry, we show that N (4, A 4 , X) X 2/3+ , and that there are N 3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S 4 , confirming a question of Gallagher. Unconditionally we have N (4, A 4 , X) X 5/6+ , and that the 2-class groups of almost all Abelian cubic fields k have size D 1/3+ k . The proofs depend on counting integral points on elliptic fibrations.

Progress towards countingD5quintic fields

“…In Section 6, we further provide numerical evidence that N (5, D 5 , X) X 2 3 +α for very small α; in particular the exponent appears to be much lower than 3 4 . Before we prove Theorem 1.1, we show that earlier work of Wong [9] in the case of G = A 4 can be handled in a similar fashion. Namely, we give a shorter proof of the following theorem: Theorem 1.2 (Wong).…”

“…In this section, we sketch a simplified (although essentially equivalent) version of Wong's proof [9] that N (4, A 4 , X) X 5 6 + as motivation for our main theorem. In this section, we assume that the reader is familiar with the arguments in Wong's paper.…”

Untitled

“…Before we prove Theorem 1.1, we show that earlier results from [Wong 2005] in the case of G D A 4 can be handled in a similar fashion. Namely, we give a shorter proof of the following theorem: Theorem 1.2 (Wong).…”

“…In this section, we sketch a simplified (although essentially equivalent) version of Wong's proof [Wong 2005] that N .4; A 4 ; X / X 5 6 C as motivation for our main theorem. In this section, we assume that the reader is familiar with the arguments in Wong's paper.…”

“…In this section, we assume that the reader is familiar with the arguments in Wong's paper. As noted in the last section, it suffices to count triples .a 2 ; a 3 ; a 4 / for which ja k j X [Wong 2005]. )…”