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Thursday, May 04, 2023
4:00 PM - 5:00 PM
Linde Hall 387

Number Theory Seminar

The unipotent mixing conjecture
Philippe Michel, École Polytechnique Fédérale de Lausanne,

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.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].