UBC Number Theory Seminar: Lior Silberman

Let M=SL2(ℤ[i])∖ℍ(3) be the Bianchi orbifold, and let {fn}∞n=1⊂L2(M) be a sequence of Hecke--Maass forms on it. We should that the probability measures on M with densities |fn(x)|2 with respect to the Riemannian volume become equidistributed on M. This generalizes the case SL2(ℤ)∖ℍ(2) due to Lindenstrauss.

In the first part of the talk I will introduce the quantum unique ergodicity problem and the setup necessary to state the theorem. In the second half I will try to give ideas of the proof, including of the full result. More generally the result holds when we replace ℚ(i) with a general number field F, SL2 with the group G of norm-1 elements of a quaternion algebra over F, ℍ(3) with G(F∞)/K∞ where F∞=F⊗ℚℝ and K∞ is a maximal compact subgrouop, and SL2(ℤ[i]) with a congruence lattice.

Joint work with Z. Shem-Tov, also relying on work with A. Zaman.