Monograph

#### Degree

Doctor of Philosophy

Mathematics

Hall, Chris

#### Abstract

Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$-torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with non-trivial $p$-torsion by the $\F_p$-rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups with non-trivial $p$-torsion as well as some estimates. In addition, we show that the Jacobian groups in this family with non-trivial $p$-torsion appear with frequency close to $1/p$, provided $M$ is irreducible.

#### Summary for Lay Audience

Characterizing a mathematical property or object often proves to be an arduous task. A more manageable approach is to consider a collection of mathematical objects of certain type and then study the variation of a particular property within that collection. More precisely, one looks for the proportion of members in the collection having said property. These techniques have proved to be fruitful in arithmetic statistics and it is in this field that our problem lies.

In this work, we deal with matroids, which are combinatorial structures that were created to study the abstract properties of linear independence. Starting with a base'' regular matroid, we construct a collection of regular matroids. To each regular matroid, we associate a finite abelian group, which is, in particular, a finite set. Understanding the structure of this group is important for many areas of mathematics, yet, little is known. One natural step in this direction is to establish when the Jacobian group of a regular matroid has $p$-torsion, that is, when is its size divisible by $p$? In this thesis, we determine the proportion of members (in the collection under consideration) having Jacobian group with non-trivial $p$-torsion. We conclude that this proportion is close to $1/p$ when the base matroid is connected.