Sato-Tate Distributions of Jacobian Varieties

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The overarching goals of this project are centered around some of the primary goals of mathematics: identifying, visualizing, and describing patterns; determining what structures lead to these patterns; and making connections between distinct but related objects. The questions that the Principal Investigator (PI) will examine in this project will use theoretical concepts and techniques from a wide range of mathematical areas, including number theory, group theory, and arithmetic geometry. More specifically, this project will focus on studying patterns in the number of solutions of polynomial equations over finite fields, and how these patterns can be understood through the study of related algebraic structures called Sato-Tate groups. Through this project, the PI will contribute to the broader community with two initiatives: co-organizing a research workshop that will support participants from groups that are historically underrepresented in mathematics; and conducting research with undergraduates at her home institution. Solutions to the polynomial equations yield points on algebraic curves, and Sato-Tate groups play an important role in the study of L-polynomials of these curves. The generalized Sato-Tate conjecture predicts that the limiting distributions of coefficients of the normalized L-polynomial of a curve converge to the distributions of traces in the Sato-Tate group associated with the Jacobian of the curve. These distributions are well-understood for genus 1 and 2 curves, and there has been significant work in genus 3. The PI will expand on the literature by further developing the theory of Sato-Tate groups and distributions for higher genus curves and their Jacobian varieties. In genus greater than 3, certain degeneracies can occur for the Hodge groups, Hodge rings, and Mumford-Tate groups associated with the Jacobian varieties, and this can affect both the identity component and the component group of the Sato-Tate group. The PI will develop new techniques for determining Sato-Tate groups in these degenerate cases. Both theoretical and computational methods will be utilized in 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.