Analytic Number Theory over Function Fields

Number theory is an area of mathematics that centers on the ordinary counting numbers and their behavior when we add and multiply them. While problems in this area are often simple to state, they can be fiendishly difficult to solve. The subfield of function field number theory aims to obtain insight on these problems by considering a kind of model or parallel universe where numbers behave differently. We consider what happens when we add or multiply numbers as normal but, except, instead of carrying digits, we simply drop the excess. Certainly arithmetic is a little easier with this modified rule, but more surprisingly, some of the most important problems in number theory become easier as well, with even some of the most difficult ones becoming solvable. (Technically, we should work in binary, or any prime base, rather than our usual base 10, for this.) Alternately, we can describe this variant arithmetic as the addition or multiplication of polynomial functions in a single variable. In this setting, we can connect number-theoretic questions to geometry, by viewing the graph of the polynomial as a geometric object. In this award the PI's research uses geometric tools to solve new problems in this area. The PI's research has resolved function field analogues of classical problems in number theory, including the twin primes conjecture and Chowla's conjecture (both joint with Shusterman), cases of the Ramanujan conjecture (joint with Templier), and conjectures about moments of L-functions. In this award the PI will continue along these lines, proving additional results about the distribution of prime numbers, L-function moments, and automorphic forms, and work in further directions such as non-abelian Cohen-Lenstra heuristics. These works are all based on etale cohomology theory, where the foundational result, Deligne's Riemann Hypothesis, allows many different analytic problems (problems about proving some inequality) to be reduced to cohomology problems (problems about calculating some of the cohomology groups of a variety or sheaf). The relevant varieties are high-dimensional, and calculating the necessary cohomology groups requires techniques like vanishing cycles theory and the characteristic cycle. 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.