Arithmetic Statistics: Geometry and Analysis

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Number theory is one of the oldest branches of mathematics, and yet it continues to have more and more applications within the sciences. Arithmetic statistics is the study of number-theoretic objects in aggregates, rather than in isolation. This study takes many different forms, several of which will be pursued in this project. The general philosophy behind the project is the interplay between theory and computation in number theory. On one hand, initial understanding of the behavior of many classes of number-theoretic objects is generally driven by the study of explicit examples; however, production of a sufficient supply of examples often depends on theory that will underpin the accuracy and range of computations. This project will lead to a deeper understanding of the statistics of fundamental behaviors of numbers. The strategies and techniques that will be utilized in this project originate in a broad range of other mathematical subjects, including analysis, geometry, algebra and in some cases, statistics. The PI will study a range of problems concerning zeta functions and L-functions associated to algebraic varieties over finite fields, function fields in positive characteristic, and number fields. These include statistical problems about point counts, in both the arithmetic aspect (Sato-Tate distributions for abelian varieties over Q and other number fields) and the geometric aspect (averages over geometric objects over a fixed finite field); algorithmic determination of L-functions; and enumeration and tabulation of function fields.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.