The Frobenius action on curves and abelian varieties

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This research project aims to study arithmetic properties of geometric objects such as curves and abelian varieties defined over different types of fields such as number fields, finite fields, global function fields and their extensions. Instead of studying each individual object one at a time, the principal investigator and her collaborators will take these objects and pack them into various types of families, and then use the geometry of the spaces parameterizing these families to deduce properties of the original objects. The main question that the principal investigator and her collaborators aim to answer is to estimate the number of special objects in these families and how often or rarely they occur. These special objects present useful and important properties making them central topics of research in many areas and directions in number theory and arithmetic geometry. Some of the target results will generalize important prior work of other mathematicians. The research program will provide many projects suitable for undergraduate and graduate students research which the principal investigator will supervise.There are two main directions the principal investigator and her collaborators will pursue with the projects, namely, to study the p-divisible groups for families of high dimensional abelian varieties and to study the structure of the ideal class groups of certain families of global function fields. There are different types of families in the research projects, such as the reductions of an abelian variety defined over a global field parameterized by the places of the base field, algebraic families of abelian varieties parameterized by a Shimura variety and sets of global function fields ordered by their discriminant. Specifically, one project aims to prove the set of ordinary primes in the reduction of certain abelian varieties with nontrivial endomorphism groups has density 1. In the opposite direction, another project aims to construct infinitely many primes at which these abelian varieties admit basic reduction, generalizing the work of Elkies’ on the infinitude of supersingular primes for elliptic curves. For ideal class group, the principal investigator and her collaborators will use Galois cohomology and computational tools to predict and prove properties of the distribution of l-torsion classes for degree l extensions of the rational function field.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.