8 Mass Equidistribution for Hecke Eigenforms

X |φ(z)|2 dx dy y = 1. Zelditch [19] has shown that as λ → ∞, for a typical Maass form φ the measure μφ := |φ(z)|2 dx dy y approaches the uniform distribution measure 3 π dx dy y . This statement is referred to as “Quantum Ergodicity.” Rudnick and Sarnak [13] have conjectured that an even stronger result holds. Namely, that as λ → ∞, for every Maass form φ the measure μφ approaches the uniform distribution measure. This conjecture is referred to as “Quantum Unique Ergodicity.” Lindenstrauss [8] has made great progress towards this conjecture, showing that, for Maass cusp forms that are eigenfunctions of the Laplacian and all the Hecke operators, the only possible limiting measures are of the form 3 π c dx dy y with 0 ≤ c ≤ 1. For illuminating accounts on this conjecture we refer the reader to [7, 8, 9, 10, 11, 13, 14, 15, 18]. Here we consider a holomorphic analog of the quantum unique ergodicity conjecture. This analog is very much in the spirit of the Rudnick-Sarnak conjectures, and has been spelt out explicitly in [10, 14]. Let f be a holomorphic modular cusp form of weight k (an even integer) for SL2(Z). Associated to f we have the measure μf := y |f(z)| dx dy y2 ,