Combinatorial Aspects of Elliptic Curves

Details Combinatorial Aspects of Elliptic Curves Combinatorial Aspects of Elliptic Curves

2007-07-23

arxiv.org/abs/0707.3179v1

Seminaire Lotharingien de Combinatoire, vol. 56 (2007) Art. B56f

Mathematics

Combinatorics

29 pages, Section 2 presented at FPSAC 2006

Scientific paper

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.

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