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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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