Moduli Spaces and Galois Theory in Arithmetic Dynamics

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Arithmetic dynamics is a relatively new discipline that brings together two major areas of mathematics: number theory, traditionally the study of the integers and integer (or rational) solutions to polynomial equations, and discrete dynamical systems, where one studies the long term behavior of functions under repeated iteration. This project will further develop and combine two facets of arithmetic dynamics: one is geometric in nature, using geometric objects (moduli spaces) to classify dynamical systems with specified dynamical behaviors, and the other is algebraic, understanding algebraic symmetries (Galois theory) associated to dynamical systems. The fields of number theory and, to some extent, arithmetic dynamics have found uses in cryptography and related areas. The PI will continue outreach activities with the aim of using cryptography as a means of introducing a more general audience to interesting mathematics. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).The field of arithmetic dynamics is heavily motivated by analogies between arithmetic geometry and dynamical systems. One important such connection is that preperiodic points for endomorphisms of projective space play a role similar to torsion points on elliptic curves (or, more generally, abelian varieties). The focus of this project is to further investigate this analogy from the moduli-theoretic and Galois-theoretic perspectives. The PI has been involved with the development of the theory of moduli spaces that parametrize endomorphisms with marked preperiodic points -- analogous to classical modular curves, which parametrize elliptic curves with marked torsion points. Such moduli spaces have already played a fundamental role in progress toward the dynamical uniform boundedness conjecture of Morton and Silverman, a dynamical analogue of the Mazur-Merel strong uniform boundedness theorem for torsion points on elliptic curves. In order to make further progress on this difficult uniform boundedness problem, the PI proposes to study the geometry of dynamical moduli spaces attached to certain dynamically interesting families of functions (e.g., quadratic rational maps with a critical point of a given period). The analogy between preperiodic points and torsion points also lends itself to a dynamical analogue of Serre's open image theorem, a finite-index result for the adelic Galois representation associated to torsion points on elliptic curves. The PI proposes studying the appropriate Galois representation attached to (pre)periodic points for rational maps -- especially in the function field setting, where results will provide new insights into the geometry of dynamical moduli spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.