Related papers: Equidistribution of Eisenstein series on geodesic …

We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies…

On convex co-compact hyperbolic surfaces with Hausdorff dimension of the limit set less than 1/2, we investigate high energy behaviour of Eisenstein Series. Eisenstein Series are non-L^2 eigenfunctions of the hyperbolic Laplacian which…

We prove a variety of quantum unique ergodicity results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet $L$-function on the $1$-line. A…

We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``shrinking'' families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite…

We consider the question of Quantum Unique Ergodicity for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series…

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruences between cusp forms and Eisenstein series on the…

We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the…

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\mathrm{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance…

Given a compact segment, $\beta$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $\beta$. We prove unconditionally a sharp lower bound for Eisenstein series along…

We consider the the intersections of the complex nodal set of the analytic continuation of an eigenfunction of the Laplacian on a real analytic surface with the complexification of a geodesic. We prove that if the geodesic flow is ergodic…

Let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=\Gamma\backslash\mathbb{H}$ be the associated finite…

We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of…

We investigate the mean value of the inner product of squared $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced…

We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound…

In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the…

For convex co-compact hyperbolic manifolds $\Gamma\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $\delta_\Gamma< n/2$, we show that the high-frequency Eisenstein series associated to a point $\xi$ "at…

We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its…

We study the quantum limits of Eisenstein series off the critical line for $\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}$, where $K$ is an imaginary quadratic field of class number one. This generalises the results of Petridis,…

We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many…

The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in "Equidistribution of Eisenstein series in the level aspect", Commun. Math. Phys. 289(3), 1131-1150 (2009). We point out errors and…