Philippe Michel: The unipotent mixing conjecture

Abstract:

Let \(q\) be a prime; it is well known (perhaps due to Peter Sarnak) that as \(q \to + \infty\) the discrete horocycle of height \((1/q, \frac{a+i}{q})\), \(a = 1, \dotsc, q\) equidistribute on the modular curve. In this talk we will explain a proof of the joint equidistribution, on the product of two copies of the modular curve of this horocycle and a multiplicative shift of it, \((\frac{a+i}q,\frac{ba+i}q)\), \(a=1,\dotsc,q\)under some natural diophantine condition on \(b/q\); this is a very special case of the mixing conjecture that Venkatesh and myself formulated a few years ago. The proof uses a mixture of the theory of automorphic forms, ergodic theory and multiplicative number theory. We will also discuss more general joint equidistribution problems as well as applications to moments of L-functions. This is joint work with Valentin Blomer.