Counting $G$-extensions by discriminant

Abstract

The problem of analyzing the number of number field extensions L/K with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava, Bhargava-Shankar-Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions L/K with fixed degree, bounded relative discriminant, and specified Galois closure.

The weak form of Malle’s conjecture and solvable groups

“…Combining these bounds with Theorem 1.2 can inductively produce upper bounds for Gextensions whenever N is an abelian normal subgroup and we know bounds on the number of G{N -extensions. In certain cases, these inductive bounds will be better than the best known bounds for the group G by other methods (such as [Sch95,Dum18]). We generally expect this to happen when inertia groups IppL{Kq ď G which are "closer to the center" of G produce smaller exponents for p in the discriminant of L{K. We include some data on what these bounds look like for groups of small degree in the appendix.…”

“…Wood [Woo09] counts abelian extensions ordered by conductor, and shows some ways in which this invariant is nicer than the discriminant. Bartel-Lenstra [BL17], Dummit [Dum18], and Johnson [Joh17] continue this philosophy by studying different questions when ordering number fields by various invariants. We have made an effort to cater to this perspective, where the admissible orderings we consider are general enough to include other orderings previously considered in the literature.…”

“…In this section, we will provide data for the upper bounds of N pK, G; Xq when G Ă Sn is a transitive, solvable subgroup and n small. We will directly compare the bounds given by iteratively applying Theorem 2.6 with Minkowski's bounds on the size of the class group to the best previously known bounds, in particular the general bounds for every group given by Dummit [Dum18] and Schmidt [Sch95].…”

“…We remark that Dummit only computes the bounds for groups 9T3, 9T4, 9T5, and 9T8 in [Dum18]. Dummit's Theorem does give bounds for all proper transitive subgroups G Ă Sn which are known to be better that Schmidt's bounds if G is primitive, but it becomes computationally intensive to find a set of primary invariants in order to compute the bound (it took Dummit's code two days to produce the bounds in degrees 5, 6, 7, 8, and for just these four groups in degree 9).…”

Galois groups of random integer polynomials and van der Waerden's Conjecture

“…Better bounds on F n (G, X) than those in Theorem 19, even when n > 5, are known for many specific choices of permutation group G (see, e.g., [45,21,7,24,16,1,41,18]).…”