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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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