# Equidistribution of Shears and Applications

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Alex Kontorovich , Rutgers University, IAS
Fine Hall 601

A shear'' is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete group (possibly of infinite co-volume). In joint work with Dubi Kelmer, we prove the regularized equidistribution of shears under large translates. We give applications including to moments of GL(2) automorphic L-functions, and to effective counting of integer points on affine homogeneous varieties (in particular resolving a missing case of the Eskin-McMullen/Duke-Rudnick-Sarnak machinery). No prior knowledge of these topics will be assumed.