Let q be a power of a prime, and E be an elliptic curve defined over ℍq. Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions ℍqk 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 ℍq.
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Acknowledgements This work first appeared in the author’s Ph.D. Thesis at the University of California, San Diego, alongside the article “Combinatorial Aspects of Elliptic Curves” . The author enthusiastically thanks his advisor Adriano Garsia for his guidance and many useful conversations. Conversations with Tewodros Amdeberhan, Norman Biggs, Dino Lorenzini, Richard Stanley, and Nolan Wallach have also been very helpful, and the referees’ comments have helped to improve this exposition. The author would like to thank the NSF and the ARCS Foundation for their support during the author’s graduate school. A preprint of the present article was presented at FPSAC 2007 in Tianjin, China.
This work was partially supported by the NSF, grant DMS-0500557 during the author’s graduate school at the University of California, San Diego, and partially supported by an NSF Postdoctoral Fellowship.
- Critical group
- Elliptic curves
- Frobenius map
- Graph Laplacian