Séminaire Lotharingien de Combinatoire, B56f (2007), 31 pp.
Combinatorial Aspects of Elliptic Curves
Given an elliptic curve C, we study here
Nk = #C(Fqk),
the number of points of C over the finite field
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 Nk =
is a (q,t)-analogue of the number of spanning trees of the wheel
graph. Additionally we develop a determinantal formula for Nk,
where the eigenvalues can be explicitly written in terms of q,
N1, and roots of unity. We also discuss here a new sequence of
bivariate polynomials related to the factorization of Nk, which
we refer to as elliptic cyclotomic polynomials because of their
Received: October 27, 2006.
Revised: June 22, 2007.
Accepted: July 21, 2007.
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