Thin Counting in Moduli Spaces

Yale University, New Haven CT

Investigators

Abstract

One of the goals of number theory is to count prime numbers in sequences of integers. In the last ten to fifteen years there has been a drive to understand a more general question: Find primes in integer sequences associated to infinite groups of symmetries of finite-dimensional spaces. Formulation of this general question followed the realization that many fundamental integer sequences are connected to groups of symmetries, for example the entries of Pythagorean triples, curvatures appearing in Apollonian circle packings, and the denominators of continued fractions formed using only a finite alphabet. In many important cases, the relevant group of symmetries is 'thin,' meaning that it is sparser than expected, and hence is not subject to classical techniques. As a result, new technology that draws on many different areas of mathematics has been developed to treat questions about thin groups and their associated integer sequences. The research in this project will study primes in integer sequences that arise in the study of moduli spaces. A moduli space is a mathematical object that encodes all geometric structures of a given type on an object, up to natural identifications. Moduli spaces often have associated groups of symmetries. These are built from integers, and so invite many intriguing number theoretic questions. In fact, these symmetry groups are sometimes thin, giving surprising new applications of the theory of thin groups. The PI will study two types of counting problems. The first concerns the number fields associated to periodic orbits of the Teichmueller flow on moduli spaces of abelian differentials. The PI has recently generalized Selberg's 3/16 Theorem to higher genus moduli spaces. This result will be used to obtain a congruence count for periodic orbits of the Teichmueller flow. In addition, Selberg's Theorem will be extended to affine invariant submanifolds of moduli space. The second type of problem concerns the Markoff-Hurwitz affine variety and its automorphism group. The PI will study the prime factors of the coordinates of integer points on this variety by obtaining congruence lattice point counting estimates and using sieve methods. This requires understanding the orbits of the automorphism group on the points of the variety over finite fields, and extensions of this question to square-free moduli. Questions about the statistics of the series of permutations obtained by the actions of a fixed global automorphism on the points on the variety over various finite fields will be studied. These questions are related to surprising numerical results obtained in collaboration with undergraduate students.