Galois Structures and Arithmetic Statistics

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The study of algebraic number fields, mathematical objects obtained from the solutions of polynomials with integer coefficients, is a very active area of research in number theory. Associated to each algebraic number field, there is an invariant, the class group, that measures the complexity of the number field structure. While computation of the class group is hard in general, numerical evidence suggests that the class group is "random" when ranging over a reasonable family of number fields, and this randomness behavior is studied in a recently active area – arithmetic statistics. One way to study this random invariant is to construct a random model which contains key parameters and use the model to simulate the distributions of class groups. This project will explore various questions about the constructions of such random models and their applications. The project also provides support for an undergraduate research experience. The research in this project is at the intersection of algebraic number theory, arithmetic statistics, group theory, and probability theory. The PI and her collaborators have produced a series of works on the non-abelian Cohen--Lenstra heuristics to give reasonable conjectures to predict the distributions of the Galois groups of the maximal unramified extensions of global fields (which are non-abelian generalizations of class groups). In particular, the non-abelian random group models constructed in this work play an important role in connecting the Galois objects with their arithmetic statistics. The PI plans to continue working along this direction, which includes: deeply studying the Galois structures of unramified extensions, proving properties of the Galois structure that are implied by the new arithmetical statistical heuristics, and studying how the random group models are affected by removing some restrictions in previous work. Explicitly, this project has three key goals: 1) modify the non-abelian Cohen—Lenstra heuristics in the case when the base field contains roots of unity; 2) study how the signature of the base field affects the distribution, and 3) generalize Gerth's 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.