The critical groups of a family of graphs and elliptic curves over finite fields

Let q be a power of a prime, and E be an elliptic curve defined over Fq . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions Fqk for all k ≥ 1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius map of E over Fq .