Mass equidistribution for Hecke eigenforms

Unimodularity of zeros of period polynomials of Hecke eigenforms

“…Rankin and Swinnerton-Dyer [14] † proved that all the zeros of Eisenstein series lie on the arc δ := {e iθ : π/3 θ π/2}, and Kohnen [8] provided explicit formulas for those zeros. On the other hand, it follows from the proof of the Quantum Unique Ergodicity conjecture by Holowinsky and Soundararajan [7] that if f k is a sequence of cuspidal Hecke eigenforms (f k of weight k) and Ω is a 'nice' set, then the elements of Z(f k ) become equidistributed in F in the sense that…”

Shrinking Scale Equidistribution for Monochromatic Random Waves on Compact Manifolds

“…Rudnick and Sarnak conjecture that the stronger property of QUE is true on any compact negatively curved surface [24]. This has been shown for examples of arithmetic origin in work of Lindenstrauss [22,23], and Bourgain-Lindenstrauss [4], Jakobson [19], Holowinsky [17], and Holowinsky-Soundararajan [16]. For a general metric, work of Anantharaman [1], Anantharaman-Nonnenmacher [2], Anantharaman-Silberman [3], and Dyatlov-Jin [10] places constraints on the measures that arise as quantum limits but it remains unknown whether the uniform measure is the only possibility.…”

Quantum Variance for Eisenstein Series

“…We have dim S k (Γ) = #H k ∼ k/12 as k → ∞. QUE for the measures dµ f = y k |f (z)| 2 dµ(z) for f ∈ H k was proved by Holowinsky-Soundararajan [8].…”