Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series
Let \(d\mu(z)=dx\,dy/y^2\) denote the volume element corresponding to the hyperbolic metric on the upper half plane \(\mathcal{H}\) and let \(E(z,s)\) be the weight \(1/2\) Eisenstein series for \(\Gamma_0(4)\). Then the authors study the limiting behavior of \(d\mu_t(z)=|E(z,1/2+it)|^2\,d\mu(z)\) on compact sets as \(t\to\infty\). The similar question for weight \(0\) Eisenstein series was studied by \textit{W. Luo} and \textit{P. Sarnak} [Publ. Math., Inst. Hautes Étud. Sci. 81, 207--237 (1995; Zbl 0852.11024)]. However, this case requires new ingredients since the Fourier coefficients of the weight \(1/2\) Eisenstein series \(E(z,s)\) are themselves expressed in terms of quadratic Dirichlet \(L\)-functions. Dirichlet series given as sums of such Fourier coefficients are typically not Eulerian.NEWLINENEWLINETo adapt the method of Luo and Sarnak, the authors study the growth of a double Dirichlet series \(Z(s,w)\) which may be realized as a sum of quadratic twists of a \(\mathrm{GL}_2\) Maass form; this series arose in work of the reviewer and \textit{J. Hoffstein} [Ann. Math. (2) 142, No. 2, 385--423 (1995; Zbl 0847.11026)], following earlier related work of \textit{D. Bump} et al. [Ann. Math. (2) 131, No. 1, 53--127 (1990; Zbl 0699.10039)]. The authors derive the continuation of this series, following the method outlined in [\textit{D. Bump} et al., Bull. Am. Math. Soc., New Ser. 33, No. 2, 157--175 (1996; Zbl 0865.11043)], give an exact functional equation, and then establish an estimate on the growth of \(Z(s,w)\) for \(s,w\) on the critical line as the imaginary parts of \(s,w\) approach \(\infty\) (independently). They call this bound the convexity bound. They then show that a stronger bound, that is a subconvexity bound, would imply that for any two compact Jordan measurable sets of \(\Gamma_0(4)\backslash \mathcal{H}\), the ratio of their measures with respect to \(d\mu_t(z)\) approaches the ratio of their volumes as \(t\to \infty\). An appendix gives bounds for Mellin transforms of the products of two Whittaker functions that are needed in the proof.
scientific article; zbMATH DE number 6361763