2017/05/31 by Peter Humphries · 1 citation
Mathematics · #math.NT #msc:11F12 #msc:58J51
paper · pdf · doi:10.1007/s00208-018-1677-9
published as Mathematische Annalen 371:3-4 (2018), 1497-1543 · 36 pages. To appear in Mathematische Annalen
arxiv created 2018/04/25 · arxiv updated 2018/11/06
We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the L4-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the L4-norm of an Eisenstein series E(z,1/2+itg) restricted to compact sets is bounded by √(log tg). The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in Γ\backslash ℍ, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindelof hypothesis for Maass eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Maass eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal classes of quadratic fields.