Abelian Varieties, Hecke Orbits, and Specialization

Elliptic curves (and their generalizations called abelian varieties) are fundamental mathematical objects that are also of great importance in other fields such as cryptography and error correcting codes. There are naturally occurring geometric spaces, called Shimura varieties, whose points classify different elliptic curves (and abelian varieties). Inside these spaces are orbits, called Hecke orbits. These orbits are not like the regular periodic orbits of the planets around the sun, but are highly unpredictable and chaotic. Indeed, each orbit is conjectured to be distributed equally throughout the Shimura variety. The principal investigator and his collaborators will use techniques from various areas of mathematics, including number theory, algebraic geometry and representation theory to study several aspects of these Hecke orbits. As part of this award the PI plans to introduce undergraduates to research in mathematics and to train graduate students on topics related to the project. The specific goals of this project are to understand the characteristic zero and characteristic p interplay of isogenies and Hecke orbits, keeping in mind applications to the long standing question of finding abelian varieties not isogenous to Jacobians. The PI also plans to study just-likely and unlikely intersections in Shimura varieties within the context of Hecke orbits, and to finally make progress towards understanding the dynamics of Hecke operators on mod p Shimura varieties, in the context of the Hecke Orbit conjecture. 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.