## Abstract

For a fixed finite solvable group *G* and number field *K*, we prove an upper bound for the number of *G*-extensions *L* / *K* with restricted local behavior (at infinitely many places) and \(\mathrm{inv}(L/K)<X\) for a general invariant “\(\mathrm{inv}\)”. When the invariant is given by the discriminant for a transitive embedding of a nilpotent group \(G\subset S_n\), this realizes the upper bound given in the weak form of Malle’s conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle’s conjecture for the transitive embedding of a solvable group \(G\subset S_n\) if we assume that for each finite abelian group *A* the average size of class group torsion \(|\mathrm{Hom}(\mathrm{Cl}(L),A)|\) is smaller than \(X^{\epsilon }\) as *L* / *K* varies over certain families of extensions with \(\mathrm{inv}(L/K)<X\).

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## Acknowlegements

I would like to thank my advisor Nigel Boston, Jordan Ellenberg, Jürgen Klüners, and Melanie Matchett Wood for many helpful conversations and recommendations. I would also like to thank anonymous referees for providing additional feedback during the submission process.

### Funding

Funding was provided by National Science Foundation (Grant No. DMS-1502553).

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## Additional information

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This work was done with the support of National Science Foundation Grant DMS-1502553.

## A Appendix: Data

### A Appendix: Data

In this section, we will provide data for the upper bounds of *N*(*K*, *G*; *X*) when \(G\subset S_n\) 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 [15] and Schmidt [30].

Some families of groups are easy to produce bounds for *N*(*K*, *G*; *X*) using computations done by hand, such as \(D_n\subset S_n\) as discussed in the introduction. In general though, we can get a more complete picture by using a computer algebra program. All of the computations in this section are done using MAGMA [7].

One of the drawbacks of Dummit’s result is the computational power necessary to compute sets of primary invariants, Dummit’s data extends to transitive groups of degree 8 and then covers only four transitive groups of degree 9 because of the length of time computations were taking. If we apply the trivial bound from Minkowski to Theorem 2.6 the bulk of the computations are done by computing a normal series for *G* and looping over elements of *G* to compute *a*(*G*) and the new upper bounds, which MAGMA is able perform very quickly by comparison.

We briefly describe the code being used in this section: For each transitive group *G* in degree *d*, we iterate through the lattice of normal subgroups of *G* to produce a list of all chief series for the group (i.e., normal series of maximal length). For each chief series

we induct along the series by applying Theorem 2.6 at the bottom of the series as follows: If \(G_1\) is central, apply Theorem 2.6 with \(N=G_1\). If \(G_1\) is not central, apply Theorem 2.6 with \(N=G_i\) for *i* the maximum index for which \(G_i\) is abelian. In the first case, *N* is central so the class group contributes nothing to the upper bound. In the second case, we use Minkowski’s bound in order to give an upper bound for the contribution of the class group. We have left over a chief series for *G* / *N* which is strictly shorter, and we have MAGMA repeat this process until we reach the end of the chief series. This is done for *each* chief series, as different series give different bounds, and the program returns the minimum bound produced by one of the chief series.

We will use nTd to denote the group TranstiveGroup(n,d) in MAGMA’s database, and we will only include solvable groups. In each column, we will give the corresponding power of *X* upper bound of *N*(*K*, *G*; *X*): the “Malle” column which shows the upper bound predicted by Malle, the “new” column which shows the unconditional bounds we prove using Theorem 2.6 with Minkowski’s bounds, the “previous” column which shows the previously best known result (or “SF” if the strong form of Malle’s conjecture is proven), and the “reference” column which gives the reference for the previously best known result. We remark that Dummit’s bounds depend slightly on the field *K*, if \(X^{a}\) is the bound given over \(\mathbb {Q}\) then \(X^{a + 1 - 1/[K:\mathbb {Q}]}\) is the corresponding bound over *K*. Whenever Dummit’s bounds are the best known, we specifically include the bound over \(\mathbb {Q}\).

To preserve space, we include only those groups for which the new upper bounds produced are better than the previously known bounds. In two case, namely 8T33 and 8T34, the new bounds improve on Dummit’s upper bounds whenever \(K\ne \mathbb {Q}\), so we include these with an *. We remark that for groups where the new upper bounds are not better than the previously known bounds, it is possible for the new bounds to be extremely poor. The worst example of small degree is

which is significantly worse than Dummit’s bound of \(X^{7/3}\). This bound is so large because 8T43 has a chief series of length 7, of which most of the factors are not central. This means that the class group increases the size of the bound at more steps of the induction.

The bounds we produce also tie with the best previously known bounds in several cases, including all nilpotent groups in the regular representation [24], all \(\ell \)-groups [24], and dihedral groups \(D_p\) in both representations, all of which will be omitted from the table. Mehta [29] announced some results around the same time as this paper: he proves Malle’s predicted upper bound for 6T4 and 6T10, and although our results are better than the previously known upper bounds before Mehta’s result we do not produce Malle’s predicted upper bound. Thus we exclude 6T4 and 6T10 from the tables. Mehta also proves unconditional upper bounds for \(C_m\rtimes C_t\subset S_n\) for \(n=m,mt\) which agree with the bounds we produce. This covers the groups 7T3, 7T4, 9T3, and 9T10 in our table, and we choose to include these bounds with a \(\dagger \) to indicate the concurrent result.

Isom. to | Malle | New | Previous | References | |
---|---|---|---|---|---|

Degree 6 | |||||

6T3 | \(S_3\times C_2\) | \(X^{1/2+\epsilon }\) | \(X^{3/4+\epsilon }\) | \(X^{7/3}\) | Dummit [15] |

6T5 | \(C_3\wr C_2\) | \(X^{1/2+\epsilon }\) | \(X^{1/2+\epsilon }\) | \(X^{7/4}\) | Dummit [15] |

6T8 | \(S_4\) | \(X^{1/2+\epsilon }\) | \(X^{3/2+\epsilon }\) | \(X^2\) | Schmidt [30] |

6T9 | \(S_3\times S_3\) | \(X^{1/2+\epsilon }\) | \(X^{3/2+\epsilon }\) | \(X^2\) | Schmidt [30] |

Degree 7 | |||||

7T3\(^{\dagger }\) | \(F_{21}\) | \(X^{1/4+\epsilon }\) | \(X^{1/2+\epsilon }\) | \(X^{7/4}\) | Dummit [15] |

7T4\(^{\dagger }\) | \(F_{42}\) | \(X^{1/3+\epsilon }\) | \(X^{5/6+\epsilon }\) | \(X^2\) | Dummit [15] |

Degree 8 | |||||

8T12 | \(SL_2(\mathbb {F}_3)\) | \(X^{1/4+\epsilon }\) | \(X^{3/4+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] |

8T13 | \(A_4\times C_2\) | \(X^{1/4+\epsilon }\) | \(X^{3/4+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] |

8T14 | \(S_4\) | \(X^{1/4+\epsilon }\) | \(X^{11/8+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] |

8T23 | \(GL_2(\mathbb {F}_3)\) | \(X^{1/3+\epsilon }\) | \(X^{3/2+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] |

8T24 | \(S_4\times C_2\) | \(X^{1/2+\epsilon }\) | \(X^{9/4+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] |

8T32 | \(X^{1/2+\epsilon }\) | \(X^{5/4+\epsilon }\) | \(X^{5/2}\) | Schmidt [30] | |

8T33\(^*\) | \(C_2^2\rtimes C_6\) | \(X^{1/2+\epsilon }\) | \(X^{9/4+\epsilon }\) | \(X^2\) | Dummit [15] |

8T34\(^*\) | \(E_4^2 \rtimes D_6\) | \(X^{1/2+\epsilon }\) | \(X^{19/8+\epsilon }\) | \(X^2\) | Dummit [15] |

Degree 9 | |||||

9T3\(^{\dagger }\) | \(D_9\) | \(X^{1/4+\epsilon }\) | \(X^{3/8+\epsilon }\) | \(X^{13/6}\) | Dummit [15] |

9T5 | \(C_3^2 \rtimes C_2\) | \(X^{1/4+\epsilon }\) | \(X^{1/2+\epsilon }\) | \(X^{19/12}\) | Dummit [15] |

9T8 | \(S_3\times S_3\) | \(X^{1/3+\epsilon }\) | \(X^{1+\epsilon }\) | \(X^2\) | Dummit [15] |

9T9 | \(X^{1/4+\epsilon }\) | \(X^{3/4+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T10\(^{\dagger }\) | \(X^{1/4+\epsilon }\) | \(X^{3/4+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T11 | \(X^{1/4+\epsilon }\) | \(X^{9/8+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T12 | \(X^{1/3+\epsilon }\) | \(X^{2/3+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T13 | \(X^{1/3+\epsilon }\) | \(X^{4/3+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T14 | \(X^{1/4+\epsilon }\) | \(X^{5/4+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T15 | \(X^{1/4+\epsilon }\) | \(X^{5/4+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T16 | \(X^{1/3+\epsilon }\) | \(X^{5/3+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T18 | \(X^{1/3+\epsilon }\) | \(X^{5/2+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T20 | \(X^{1/2+\epsilon }\) | \(X^{3/2+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T21 | \(X^{1/2+\epsilon }\) | \(X^{3/2+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T22 | \(X^{1/2+\epsilon }\) | \(X^{11/6+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] | |

9T24 | \(X^{1/2+\epsilon }\) | \(X^{7/2+\epsilon }\) | \(X^{11/4}\) | Schmidt [30] |

We remark that Dummit only computes the bounds for groups 9T3, 9T4, 9T5, and 9T8 in [15]. Dummit’s Theorem does give bounds for all proper transitive subgroups \(G\subset S_n\) 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).

One should notice that our new bounds appear to improve many more results in degree 9 than in degree 8. This is for several reasons. \(|S_8|\) is divisible by a much much larger power of 2 than the power of 3 dividing \(|S_9|\), which means there are a lot more 2-groups in \(S_8\) for which (conjecturally) sharp bounds were already proven by Klüners–Malle. Because 8 is divisible by 2, \(S_8\) also has some transitive subgroups of the form \(C_2\wr H\) for which the strong form of Malle’s conjecture holds [23], while \(S_9\) has no such subgroups. Lastly, 8 has more divisors than 9, which means there are a lot more ways to make transitive subgroups with longer normal series and we know that that the unconditional bounds coming from Theorem 2.6 with Minkowski’s bounds get worse for longer normal series.

We would expect this pattern to continue to hold for larger degrees. Our new results are likely to improve the best known upper bounds for more groups in odd degrees with fewer divisors.

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Alberts, B. The weak form of Malle’s conjecture and solvable groups.
*Res. number theory* **6, **10 (2020). https://doi.org/10.1007/s40993-019-0185-7

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### Keywords

- Malle’s conjecture
- Solvable
- counting

### Mathematics Subject Classification

- 11R21
- 11R37