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Quantum ergodicity of eigenfunctions on $$\text{PSL}_ 2(\mathbb{Z}) \backslash H^ 2$$. (English) Zbl 0852.11024
The quantum ergodicity in the title concerns distribution results for measures $$|f(z) |^2 d\mu(z)$$ on the upper half plane $$H^2$$ modulo the full modular group $$\Gamma= \text{PSL}_2 (\mathbb{Z})$$, where $$f$$ is either a Maass cusp form, or an Eisenstein series with its spectral parameter on the critical line. $$d\mu (z)$$ denotes the invariant measure $$y^{-2} dx dy$$.
Let $$u_1, u_2, \dots$$ be a maximal orthogonal system of Maass cusp forms, with corresponding eigenvalues $$\lambda_1, \lambda_2, \dots\;$$. It is shown that for smooth integrable functions $$F$$ on $$X= \Gamma \setminus H^2$$: $\sum_{\lambda_j\leq x} \;\Biggl|\int_X F|u_j |^2 d\mu- {\textstyle {3\over \pi}} \int_X F d\mu \Biggr|^2 \ll_\varepsilon C_F x^{1/2+ \varepsilon} \qquad (x\to \infty)$ for each $$\varepsilon> 0$$. The constant $$C_F$$ depends on supremum norms of partial derivatives of $$F$$ up to order 8.
Let $$E(z, s)$$ be the Eisenstein series of weight zero, with eigenvalue $$s- s^2$$. As $$X$$ has infinite mass for the measure $$|E(z, 1/2+ it)|^2 d\mu (z)$$, the quantum ergodicity is formulated in terms of compact Jordan measurable subsets of $$X$$. It is shown that for each such subset $$A$$: $\int_A |E(z, 1/2+ it)|^2 d\mu (z)\sim {\textstyle {48 \over \pi}} \mu(A) \log t \qquad (t\to \infty).$ Other results in the paper are an improvement in the error term of the prime geodesic theorem for $$X$$, and the estimate $\sum_{\lambda_j\leq x} \sup_B \Biggl|\int_B |u_j (z) |^2 d\mu (z)- {\textstyle {3\over \pi}} \mu (B) \Biggr|^2 \ll x^{20/ 21+ \varepsilon} \qquad (x\to \infty).$ $$B \subset X$$ runs over the injective geodesic circles in $$X$$.
The proofs use many techniques and results of the theory of real analytic modular forms: $$L$$-functions associated to modular forms, approximation by Poincaré series, Hecke operators, the Kuznetsov sum formula, and bounds on Fourier coefficients of Maass cusp forms.

##### MSC:
 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F30 Fourier coefficients of automorphic forms
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