Quantum Unique Ergodicity for half-integral weight automorphic forms
We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for Γ₀(4) lying in Kohnen's plus subspace and for half-integral weight Hecke Maaβ cusp forms for Γ₀(4) lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on Γ₀(4)∖H as the weight tends to infinity.
© 2020 Duke University Press. Received: 4 September 2017; Revised: 11 June 2019; First available in Project Euclid: 29 January 2020. This work is an outgrowth of our joint research with Kaisa Matomäki , and we are especially thankful to her for her numerous ideas and suggestions. We are also very grateful to Zeév Rudnick and Peter Sarnak for many conversations related to this project and for their encouragement. We would also like to thank Kannan Soundararajan for many discussions on moments and Valentin Blomer for very useful pointers to the literature on Whittaker functions. Finally, we are also grateful to the anonymous referees for a very careful reading of the paper. Radziwiłl's work was partially supported by a Sloan Fellowship.
Submitted - 1606.04119.pdf