Research Interests

Number Theory
Algebraic Dynamics
Integral Lattices

Algebraic Dynamics

Whenever we are given a function f from a set X to itself, we may view the function as generating a dynamical system by considering the effect of repeatedly applying the function f to the set X. One way to understand the resulting system is to ask what can be said about the orbit
{x, f(x), f 2(x), f 3(x),...}
of a point x in X, where f k denotes f iterated k times. In this general scenario, we may consider the points x that are fixed by f, for which f(x) = x, the points x that are periodic under f, for which f k(x) = x for some positive k, and the points x that are preperiodic under f, for which f m(x) is periodic for some m (these are the points with finite orbit). To formulate more sophisticated questions about the orbits of points, we need to impose some added structure on the set X. In the classical study of dynamical systems, this structure is topological or analytic in nature, and one finds that the periodic and preperiodic points of f govern the orbits of the other points in X. For instance, we may set X to be the Riemann sphere P1(C) and require f to be a holomorphic function. An illustration of the role of periodic points is provided by the special case where f is a polynomial. In this case, is an attracting fixed point, which means that for every point z in some neighborhood of , f k(z) approaches as k goes to . Those points whose orbits are not attracted to form the filled Julia set of f, a set whose shape is determined by the attracting or repelling nature of the preperiodic points of f in C.

In algebraic dynamics, the structure with which the underlying set and function in our dynamical system are endowed comes from the realm of algebraic geometry. In this case, the function will be a morphism of an algebraic variety over a field which may have no inherent topological structure. If the base field is Q or a finite extension of Q, then the study of the algebraic dynamics takes on a number-theoretic aspect. Much of the theory can be developed by studying algebraic dynamical systems on the projective space Pn(K), where K is a number field. In the one-dimensional case, this is a dynamical system defined by iteration of a rational function f in K(z) acting on the projective line P1(K) = K {}. When f is defined over a number field, the set Preper(f) of preperiodic points of f is an intriguing object of study. There is an outstanding uniform boundedness conjecture, due to Morton and Silverman, which in the one-dimensional case states:
For every D 1 and every d 2, there is a constant C (D, d) such that for every number field K with [K:Q] D and every f in K(z) of degree d, #PreperK(f) C (D, d).
In the simplest case, where D = 1 and d = 2, the conjecture would imply the existence of a bound C so that every function f in Q(z) that has degree 2 has at most C preperiodic points in Q. The full Morton-Silverman conjecture, applied to projective spaces of every dimension, is known to imply the uniform boundedness conjecture for torsion of abelian varieties. At the moment, it is not known whether the conjecture holds even in the case where f is a quadratic polynomial.

Gregory Call and I have given a non-uniform bound for the number of preperiodic points over Q of a quadratic polynomial. To my knowledge, it is the best bound known to date. To construct our bound, we use the global canonical height associated to f, as described by Call and Silverman, which is zero precisely at the preperiodic points of f. This global canonical height decomposes into a sum of local canonical heights defined at each of the places of Q, and the local canonical heights reflect the local dynamics over Cp (the smallest complete, algebraically closed field containing Qp) and over C. In the polynomial case, each local canoncial height is non-negative, and so a point in P1(Q) is preperiodic if and only if all of the local canonical heights are zero. The result is a bound on the standard absolute value of the point and a set of congruence conditions that bound #PreperQ(f) in terms of the number of primes in the coefficients of f. Our techniques have been extended to families of higher-degree polynomials over higher number fields by Call and others.

I am working with Robert Benedetto on extending the local-height techniques in the polynomial case together with results on p-adic Fatou sets due to Benedetto and Juan Rivera-Letelier to non-polynomial functions with ramified fixed points. In dynamical terms, the function f is polynomial if and only if it has a totally ramified fixed point at . In this case, the set of points with non-zero p-adic local height is precisely the analytic component of the Fatou set containing as defined by Benedetto. If is merely ramified (i.e., superattracting), it is still possible to apply a modification of the polynomial-case analysis to the dynamics of f.

A good basic overview of height functions on Pn(K) is given in Silverman's Arithmetic of Elliptic Curves. The Call/Goldstine paper is:

Canonical Heights on Projective Space, Journal of Number Theory 63 (1997), no. 2, 211 - 243.

The relevant p-adic dynamics can be found in recent papers of Benedetto and Rivera-Letelier.

Integral Lattices

Many interesting lattice constructions arise from representation theory. In 1976, Thompson wrote a two-page paper entitled Finite Groups and Even Lattices. The paper begins, "I would like to record a consequence of what appears to be a rare occurence," and consists entirely of the following result:
Suppose G is a finite group and M is a finitely generated torsion-free ZG-module such that for each prime p, M/pM is irreducible. Then, either M = Z or there is a G-admissible positive definite integral inner product on M that is unimodular and even.

In his paper Group Representations and Lattices, Gross describes the notion of a globally irreducible representation of a group G. In the event that the representation of G is absolutely irreducible, this global irreducibility coincides with the criterion of Thompson. There are many examples of globally irreducible representations, and the principle has been applied and amplified in various papers of Tiep. Among these is Basic spin representations of 2Sn and 2An as globally irreducible representations, which extends previous results of Gow on spin representations of the double covers of Sn and of An.

My work in this area deals with spin representations associated to the algebraic group Spin(L) of a suitable lattice L. In this case, the analog of global irreducibility is provided by the Lie group structure of Spin(L). The corresponding Lie algebra, along with its localizations, plays a key role. The basic construction is outlined in the preprint of Spin Representations and Lattices.

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