Zeros of Modular Forms in Thin Sets and Effective Quantum Unique Ergodicity
نویسندگان
چکیده
We study the distribution of zeros of holomorphic Hecke cusp forms in several “thin” sets as the weight, k, tends to infinity. We obtain unconditional results for slowly shrinking (with k) hyperbolic balls. This relies on a new, effective, proof of Rudnick’s theorem and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. In addition, assuming the Generalized Lindelöf Hypothesis or looking only at almost all Hecke cusp forms, we get a formula with a power saving bound for the error term. We also study the zeros high up in the cusp. Here it is conjectured by Ghosh and Sarnak that these zeros lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high up in the cusp are on these geodesics. For all forms, assuming the Generalized Lindelöf Hypothesis, we obtain lower bounds on the number of such zeros which are significantly better than the unconditional results.
منابع مشابه
The Quantum Unique Ergodicity Conjecture for Thin Sets
We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or “shrinking” families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite geodesic connecting 0 and ∞ inside the modular surface.
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