Math Table/Open Neighborhood Seminar: Class numbers, prime geodesics, and automorphic forms (after Sarnak)

MATH TABLE

Speaker:

Kenz Kallal - Harvard Undergraduate

Given fixed integers a, b, c, which primes can be written as aX^2 + bXY + cY^2 for integers X, Y? This simple question in number theory has generated vast amounts of mathematics over the past 400 years. Of central importance to the answer in the general case are more abstract quantities called class numbers. These are individually very mysterious, but on average seem to be well-behaved. This talk is about an asymptotic law for truncated averages of class numbers, originally from Sarnak's 1980 Stanford thesis. The proof will take us far away from the elementary terms in which the original question was stated: first to the Riemannian geometry of modular curves, and then (via the Selberg trace formula) to the analytic theory of automorphic forms.

Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09