The critical groups of a family of graphs and elliptic curves over finite fields
Gregg Musiker
DOI: 10.1007/s10801-008-0162-z
Abstract
Let q be a power of a prime, and E be an elliptic curve defined over \mathbb F q \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 \mathbb{F}_{q^{k}} for all k\geq 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 \mathbb{F}_{q} .
Pages: 255–276
Keywords: keywords elliptic curves; critical group; graph Laplacian; Frobenius map
Full Text: PDF
References
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2. Biggs, N.L.: Chip-firing and the critical group of a graph. J. Algebr. Comb. 9, 22-45 (1999)
3. Biggs, N.L.: The critical group from a cryptographic perspective. Bull. London Math. Soc. (2007). 8 pages
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5. Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613- 1616 (1990)
6. Gabrielov, A.: Abelian avalanches and Tutte polynomials. Physica A 195, 253-274 (1993)
7. Jacobson, B., Neidermaier, A., Reiner, V.: Critical groups for complete multipartite graphs and Cartesian products of complete graphs (2002).
8. Lenstra, H.W.: Complex multiplication structure of elliptic curves. J. Number Theory 56, 227-241 (1996)
9. Lorenzini, D.: On a finite group associated to the Laplacian of a graph. Discr. Math. 91, 277-282 (1991)
10. Lorenzini, D.: Smith normal form and Laplacians (2007).
11. Maxwell, M.: Enumerating bases of self-dual matroids (2006). J Algebr Comb (2009) 30: 255-276
12. Musiker, G.: Combinatorial aspects of elliptic curves. Seminaire Lotharingien de Combinatoire 56 (2007), Article B56f
13. Postnikov, A., Shapiro, B.: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Amer. Math. Soc. 356(8), 3109-3142 (2004)
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15. Stanley, R.P.: Enumerative combinatorics. Cambridge Studies in Advanced Mathematics, vol. 62.