** The Weak Form of Malle’s Conjecture and Solvable Groups** (arxiv preprint)

Research and Number Theory **6**, 10 (2020) doi:10.1007/s40993-019-0185-7

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 $${\rm inv}(L/K)< X$$ for a general invariant “$${\rm inv}$$”. 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 in 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 $$|{\rm Hom({\rm Cl}(L),A)}|$$ is smaller than $$X^{\epsilon}$$ as $$L/K$$ varies over certain families of extensions with $${\rm inv}(L/K)<X$$.

** Statistics of the First Galois Cohomology Group: A Refinement of Malle’s Conjecture **(2019)

submitted for publication (arxiv preprint)

Malle proposed a conjecture for counting the number of $$G$$-extensions $$L/K$$ with discriminant bounded above by $$X$$, denoted $$N(K,G;X)$$, where $$G$$ is a fixed transitive subgroup $$G\subset S_n$$ and $$X$$ tends towards infinity. We introduce a refinement of Malle’s conjecture, if $$G$$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $$Z^1(K,G)$$ (or equivalently $$1$$-coclasses in $$H^1(K,G)$$) with bounded discriminant. This has a natural interpretation given by counting $$G$$-extensions $$F/L$$ for some fixed $$L$$ and prescribed extension class $$F/L/K$$.

If $$T$$ is an abelian group with any Galois action, we compute the asymptotic main term of this refined counting function for $$Z^1(K,T)$$ (and equivalently for $$H^1(K,G)$$) and show that it is a natural generalization of Malle’s conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $$G\subset S_n$$ over $$K$$ and $$G$$ has an abelian normal subgroup $$T\le G$$ we prove a nontrivial lower bound for $$N(K,G;X)$$ given by a nonzero power of $$X$$ times a power of $$\log X$$. For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle’s predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Klüners’ counter example to Malle’s conjecture and verify the corrected lower bounds predicted by Turkelli.

** Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations **(2017)

We study solutions to the Brauer embedding problem with restricted ramification. Suppose $$G$$ and $$A$$ are a abelian groups, $$E$$ is a central extension of $$G$$ by $$A$$, and $$f:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow G$$ a continuous homomorphism. We determine conditions on the discriminant of $$f$$ that are equivalent to the existence of an unramified lift $$\widetilde{f}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow E$$ of $$f$$.

As a consequence of this result, we use conditions on the discriminant of $$K$$ for $$K/\mathbb{Q}$$ abelian to classify and count unramified nonabelian extensions $$L/K$$ normal over $$\mathbb{Q}$$ where the (nontrivial) commutator subgroup of $${\rm Gal}(L/Q)$$ is contained in its center. This generalizes a result due to Lemmermeyer, which states that a quadratic field $$\mathbb{Q}(\sqrt{d})$$ has an unramified extension normal over $$\mathbb{Q}$$ with Galois group $$H_8$$ the quaternion group if and only if the discriminant factors $$d=d_1d_2d_3$$ as a product of three coprime discriminants, at most one of which is negative, satisfying the condition $$\left(\frac{d_i d_j}{p_k}\right)$$ on Legendre symbols for each $$\{i,j,k\}=\{1,2,3\}$$ and $$p_k$$ a prime dividing $$d_k$$.

** The distribution of $$H_8$$-extensions of quadratic fields** (2016) joint with Jack Klys

to appear in IMRN (arxiv preprint)

We compute all the moments of a normalization of the function which counts unramified $$H_8$$-extensions of quadratic fields, where $$H_8$$ is the quaternion group of order 8, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified $$G$$-extensions of quadratic fields for $$G$$ in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $$H_8$$-extensions.

** Cohen-Lenstra Moments for Some Nonabelian Groups** (2016)

submitted for publication (arxiv preprint)

Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field $$k$$ to a fixed odd abelian group is 1. Class field theory tells us that the class group is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of $$k$$, so we could equivalently say the expected number of unramified G-extensions of $$k$$ is $$1/{\#}{\rm Aut}(G)$$ for a fixed abelian group $$G$$. We generalize this to asking for the expected number of unramified $$G$$-extensions Galois over $$k$$ for a fixed finite group $$G$$, with no restrictions placed on $$G$$. We review cases where the answer is known or conjectured by Boston-Wood, Boston-Bush-Hajir, and Bhargava, then answer this question in several new cases. In particular, we show when the expected number is zero and give a nontrivial family of groups realizing this. Additionally, we prove the expected number for the quaternion group $$Q_8$$ and dihedral group $$D_4$$ of order 8 is infinite. Lastly, we discuss the special case of groups generated by elements of order 2 and give an argument for an infinite expected number based on Malle’s conjecture.

*Dissertation*

My PhD thesis is based on the work on the four papers that I wrote during my time in graduate school from 2013-2018. I reformatted this material for my dissertation to have consistent notation, narrative, and a more detailed introduction.

Dissertation: Galois Cohomology and Number Field Counting

Presentation: Slides