Combinatorial Aspects of Elliptic Curves

TL;DR

This paper explores the combinatorial properties of the sequence counting points on elliptic curves over finite fields, revealing new formulas, polynomial sequences, and connections to graph theory.

Contribution

It introduces a novel combinatorial interpretation of point counts on elliptic curves, including a determinantal formula and elliptic cyclotomic polynomials.

Findings

N_k expressed via a (q,t)-analogue of wheel graph spanning trees

Determinantal formula with explicit eigenvalues

Introduction of elliptic cyclotomic polynomials

Abstract

Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor in addition to the usual number theoretical interpretations. In particular we prove that $N_k = - W_k(q, - N_1)$ where W_k(q,t) is a (q,t)-analogue of the number of spanning trees of the wheel graph. Additionally we develop a determinantal formula for N_k where the eigenvalues can be explicitly written in terms of q, N_1, and roots of unity. We also discuss here a new sequence of bivariate polynomials related to the factorization of N_k, which we refer to as elliptic cyclotomic polynomials because of their various properties.