Archive | 2019
Some applications of rigidity of higher rank diagonalizable actions to number theory
Abstract
In the last 40 years, homogeneous dynamics has proven a very potent toolbox to attack problems of number theoretic flavor. A huge stepping stone was Ratner’s proof of Raghunathan’s conjectures, proving that unipotent flows are rigid, which was followed by several interesting applications in number theory. Similarly, diagonalizable flows naturally occur in number theory. However, unlike unipotent flows, they are not rigid and in general can have all kinds of orbit closures, as can be illustrated by the geodesic flow on the unit tangent bundle to the modular surface. However, the situation is quite different for higher rank flows or actions. This thesis presents three applications of rigidity of higher rank diagonalizable actions to problems of number theoretic flavor. In the first chapter, we give a short introduction to the context as well as a description of the problems described below. In the second chapter, we use spectral gap and a higher rank diagonalizable action in order to prove effective equidistribution of primitive rational points along horospheres in Hilbert modular surfaces. In the third chapter, in joint work with Manfred Einsiedler and Nimish Shah we again use spectral gap and a higher rank diagonalizable action in order to prove effective equidistribution of primitive rational points along horocycles in the modular surface. Moreover, we use entropy to prove a disjointness result for higher rank actions on tori and the modular surface. From this we deduce joint equidistribution of primitive rational points in the modular surface and of conjugate tuples of primitive rational points in the two-torus, i.e. disjointness to Kloosterman sums. In the fourth chapter, in joint work with Menny Aka, Philippe Michel, and Andreas Wieser we use a joinings classification for certain higher rank diagonalizable actions by Manfred Einsiedler and Elon Lindenstrauss to prove equidistribution of joint reductions of complex multiplication elliptic curves among the super singular curves for several distinct primes as the discriminant tends towards infinity in absolute value along a congruence condition.