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

Authors:
Gregg Musiker
Affiliations:
Mathematics Department, Massachusetts Institute of Technology, Cambridge, USA 02139
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2009

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Abstract

Let q be a power of a prime, and E be an elliptic curve defined over $\mathbb{F}_{q}$ . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions $\mathbb{F}_{q^{k}}$ for all k驴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 $\mathbb{F}_{q}$ .