Geodesic
The critical groups of a family of graphs and elliptic curves over finite fields
Journal of Algebraic Combinatorics, Tome 30 (2009) no. 2, pp. 255-276.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $q$ be a power of a prime, and $E$ be an elliptic curve defined over $\mathbb F _{ q}$ mathbbF_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 mathbbF_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 $\mathbb F _{ q}$ mathbbF_q .
Keywords: keywords elliptic curves, critical group, graph Laplacian, Frobenius map 
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     author = {Musiker, Gregg},
     title = {The critical groups of a family of graphs and elliptic curves over finite fields},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2009__30_2_a0/}
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Musiker, Gregg. The critical groups of a family of graphs and elliptic curves over finite fields. Journal of Algebraic Combinatorics, Tome 30 (2009) no. 2, pp. 255-276. http://geodesic.mathdoc.fr/item/JAC_2009__30_2_a0/