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Séminaire Lotharingien de Combinatoire, B56f (2007), 31 pp.

# Gregg Musiker

# Combinatorial Aspects of Elliptic Curves

**Abstract.**
Given an elliptic curve *C*, we study here
*N*_{k} = #*C*(**F**_{qk}),
the number of points of *C* over the finite field
**F**_{qk}. 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.

Received: October 27, 2006.
Revised: June 22, 2007.
Accepted: July 21, 2007.

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