A dissertation submitted in partial satisfaction of the requirements for the degree Doctor

OF THE DISSERTATION A Combinatorial Comparison of Elliptic Curves and Critical Groups of Graphs by Gregg Joseph Musiker Doctor of Philosophy in Mathematics University of California San Diego, 2007 Professor Adriano Garsia, Chair In this thesis, we explore elliptic curves from a combinatorial viewpoint. Given an elliptic curve E, we study here Nk = #E(Fqk), the number of points of E over the finite field Fqk . 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 Nk = −Wk(q, t)|t=−N1 where Wk(q, t) is a (q, t)-analogue for 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 various properties. The above formula for Nk in terms of Wk motivates a closer examination of the relationship between points on an elliptic curve E over Fqk and spanning trees on the wheel graph Wk. An elliptic curve E has an abelian group structure, and indeed the set of spanning trees of a graph also has an abelian group structure. Here we study one isomorphic to the critical group of the graph, which has ties to the theory of chip-firing games and abelian sandpile models of dynamical systems. While we first focus on the relationship between the integer sequences {Nk} and {Wk(q,N1)}, we also compare these two group structures, illustrating that the xi connections between elliptic curves and spanning trees run even deeper. Numerous theorems which are true for elliptic curve groups have analogues in terms of critical groups of the (q, t)-wheel graph. Additionally, the theory of critical groups will also allow us to re-interpret the group elements as the set of admissible words for a primitive circuit in a specific deterministic finite automaton. As an application, we will then compare the zeta function of an elliptic curve and the zeta function of the corresponding cyclic language.