Distribution of the p-Torsion of Jacobian Groups of Regular Matroids
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.