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

摘要

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 asequence of polynomials which express the N_k's, the number of points on E overthe 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 acombinatorial interpretation in terms of spanning trees of wheel graphs. Inthis sequel, we explore further ramifications of this connection. Inparticular, we highlight a relationship between elliptic curves and chip-firinggames on graphs by comparing the groups structures of both. As a coda, weconstruct a cyclic rational language whose zeta function is dual to that of anelliptic curve.