Arithmetic Intersection on Shimura Varieties and Properties of Abelian Varieties

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This research project concerns work in arithmetic geometry, a branch of mathematics that studies polynomial equations over the integers. Such equations define geometric objects; among them the most accessible and fundamental ones are elliptic curves (one-dimensional case) and abelian varieties (higher-dimensional analogue). A modern way to approach these objects is to study them in families, and certain geometric objects, so-called Shimura varieties, have been used to parametrize these families. Arithmetic properties of abelian varieties can be translated into arithmetic-geometric properties of the corresponding Shimura variety. In this way, the principal investigator and her collaborators will be able to use methods from different areas of mathematics such as algebraic geometry, number theory, and representation theory to study the arithmetic of abelian varieties.The main theme of this project is the infinitude of certain thin sets of primes arising from reduction types of abelian varieties. When the abelian varieties are over number fields, the principal investigator and her collaborators are aiming for results along the line of Elkies' theorem on supersingular reductions of elliptic curves. The research will focus on nonsimple reductions or reductions with higher Picard rank and establish a general framework to treat certain abelian varieties of arbitrarily high dimension. In the function field case, certain new phenomena appear and are related to the geometry of the Newton strata of the corresponding Shimura variety. The philosophy of this project is related to the Kudla program on arithmetic intersection of special cycles on Shimura varieties. In addition to the usual setting of the Kudla program, certain non-special cycles are studied in the framework of this project.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.