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arXiv · 0710.0574

Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs

Gregg Musiker
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Abstract

Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves and chip-firing games on graphs by comparing the groups structures of both. As a coda, we construct a cyclic rational language whose zeta function is dual to that of an elliptic curve.

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Gregg Musiker. 2007-10-02. Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs. https://arxiv.org/abs/0710.0574

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