Brett Nasserden, Department of Pure Mathematics, University of Waterloo
"Applications of the minimal model program in arithmetic dynamics"
Arithmetic dynamics is a rapidly growing field of mathematics that lies in the intersection of algebraic geometry, number theory and complex dynamics. Many interesting number theoretic problems admit a dynamical generalization. Understanding this dynamical generalization can shed new light on the original problem while also leading to new and interesting challenges in arithmetic and geometry.
One of the prominent outstanding problems in higher dimensional arithmetic dynamics is the Kawaguchi-Silverman conjecture. Given a surjective morphism f:X-> X there is a number called the dynamical degree of f which measures the dynamical complexity of f; the dynamical degree is a geometric measure of the complexity of f. On the other hand, given a point P in X there is an arithmetic notion of the complexity of the forward orbit of P under f, called the arithmetic degree. The Kawaguchi-Silverman conjecture predicts that these two measures of complexity coincide when P has a Zariski dense forward orbit.
A class of varieties for which the Kawaguchi-Silverman conjecture remains open are projective bundles over algebraic curves. Work by Satriano and Lesieutre has shown that the last remaining open cases are the projective bundles associated to semi-stable degree 0 vector bundles on elliptic curves. We take up this case and prove the Kawaguchi-Silverman conjecture for a large class of these projective bundles.
Recently, a generalization of the Kawaguchi-Silverman conjecture called the sAND conjecture has been proposed. The sAND conjecture involves the studying the set of points with small arithmetic degree. To study this set, it is important to understand which numbers can occur as a small arithmetic degree. In the second part of this thesis, we begin this study. Kawaguchi and Silverman showed that the set of possible arithmetic degrees of a morphism is given by the magnitude of eigenvalues of a linear pull back mapping on cohomology. We ask, is every such eigenvalue necessarily an arithmetic degree? We show that this question has a negative answer for Abelian varieties and a positive answer for simplicial toric varieties.
Please contact Nancy Maloney (nfmalone@uwaterloo.ca) if you are interested in virtually attending Brett's talk.