Symmetries and Statistics in Arithmetic

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Number fields, algebraic structures associated with the roots of polynomials, are important objects in algebraic number theory. A crucial component in studying these algebraic structures is to understand interesting invariants of it, and the symmetry of the number field heavily influences the behavior of these invariants. More precisely, although the invariants can be mysterious and random for a single number field, as in a family, their statistics obey certain distributions determined by this symmetry. The goal of this project is to study these invariants utilizing their symmetry from a statistical point of view. In particular, this research will prove new results on invariants for number fields with special symmetry, discover new phenomena in arithmetic structures motivated by statistical results, and propose new statistical measurements of these invariants. This project will also include research opportunities for graduates, undergraduates and postdocs, and will facilitate in-depth conversations between the analytic and algebraic sides of number theory. More concretely, this project will investigate distribution results for discriminants and class groups of global fields with a fixed Galois group, using tools from algebraic number theory and analytic number theory. For the distribution of discriminants, this project will prove new asymptotic distribution, upper bound and lower bound using relative invariants and inductive ideas. It will also include discussions of global fields with different characters and symmetry. For the distribution of class groups, this project will study both extremal behavior for a single number field and statistical behavior for a family of number fields. It includes investigating new existence-type arithmetic questions and representation theory questions suggested by distribution problems. On the other hand, this project will propose and study generalizations and variations of the class number problems. Finally, this project will also address fundamental questions on the distribution of roots of polynomials. This includes giving sharp inequalities leading to equidistribution results and constructing optimal root distributions with respect to various functionals.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.