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

TL;DR

This paper explores the deep combinatorial connections between elliptic curves over finite fields, spanning trees of wheel graphs, and chip-firing games, revealing new insights into their algebraic and combinatorial structures.

Contribution

It establishes a novel relationship between elliptic curves and chip-firing games on graphs by comparing their group structures, extending previous combinatorial interpretations.

Findings

Polynomials expressing point counts on elliptic curves relate to spanning trees of wheel graphs.

A connection between elliptic curves and chip-firing game group structures is demonstrated.

A cyclic rational language with a zeta function dual to that of an elliptic curve is constructed.

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.