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Estimates for Rankin-Selberg $$L$$-functions and quantum unique ergodicity. (English) Zbl 1006.11022
This is the second of a series of papers which are concerned with proving new estimates for $$L$$-functions on the critical line. The usual approach to this problem starts from the trivial bounds in the domains $$\operatorname{Re} s>1$$ and $$\operatorname{Re} s<0$$ and uses the Phragmén-Lindelöf interpolation method. This method gives the so-called convexity bound for $$L(\frac{1}{2}+ it,f)$$, which however turns out to be too weak for certain applications. Sharper estimates are known for $$L$$-functions of degree one (Dirichlet $$L$$-functions) and degree two. For general $$L$$-functions of higher degree, the deduction of bounds better than the convexity bound is an important open problem. In the present work the author establishes such a bound (with respect to the weight $$k$$ of $$f$$) for a class of special degree four $$L$$-functions, that is for Rankin-Selberg $$L$$-functions of two degree two $$L$$-functions (one of which – $$g$$ – is kept fixed) for holomorphic cusp forms $$f$$ and $$g$$ on $$\Gamma_0(N)$$. The result is applied to the problem of so-called quantum unique ergodicity: Let $$f$$ be a Hecke cusp form on $$\Gamma_0(N)$$ such that
(i) $$f$$ is a holomorphic cusp form of even integral weight $$k$$, or
(ii) $$f$$ is a Maaß cusp form with eigenvalue $$\lambda$$,
and consider the measure (normalized to be a probability measure) $d\mu_f:= y^k|f(z)|^2 \frac{dx dy}{y^2}$ on $$\Gamma_0(N) \setminus \mathbb{H}$$. Then the equidistribution conjecture of Rudnik and Sarnak claims: As $$k\to \infty$$ in the holomorphic case or $$\lambda\to \infty$$ in the case of Maaß forms, we have $d\mu_f\to \frac{1} {\text{vol} (\Gamma_0(N) \setminus \mathbb{H})} \frac{dx dy} {y^2}$ (in the sense of weak convergence). As an application of his subconvexity, bound the author shows: The equidistribution conjecture is true for holomorphic CM forms.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 81Q50 Quantum chaos 11M41 Other Dirichlet series and zeta functions 11F11 Holomorphic modular forms of integral weight 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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