Number Theory Seminar

Let $q$ be a prime; it is well known (due to Peter Sarnak) that as $q\rightarrow+\infty$ the discrete horocycle of height $1/q$, $\frac{a+i}q,\ a=1,\cdots,q$ equidistribute on the modular curve.

Here we consider the joint equidistribution of this horocycle and a multiplicative shift of it,
$(\frac{a+i}q,\frac{ba+i}q)\ a=1,\cdots,q$ on the product of two modular curves.

We prove the joint equidistribution of these pairs under some natural diophantine condition on $b/q$; this is a special case of the mixing conjecture that Venkatesh and myself formulated a few years ago. The proof is a mixture of the theory of automorphic forms, of ergodic theory and multiplicative number theory. This is joint work with Valentin Blomer.