Related papers: New variants of arithmetic quantum ergodicity

The possibility of deformation of two body quantum Calogero-Moser-Sutherland models is studied. Obtained are some necessary conditions for the singular locus of the potential function. Such locus is determined if it consists of two, three…

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the…

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This…

We apply techniques from harmonic analysis to study the $L^p$ norms of Maass forms of varying level on a quaternion division algebra. Our first result gives a candidate for the local bound for the sup norm in terms of the level, which is…

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\Q(\sqrt{5})$. In those…

We develop a $GL_{qp}(2)$ invariant differential calculus on a two-dimensional noncommutative quantum space. Here the co-ordinate space for the exterior quantum plane is spanned by the differentials that are commutative (bosonic) in nature.

Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions…

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on $\Gamma_0(D) \backslash \mathbb H$ in the large eigenvalue limit, for certain fixed $D$. As predicted in the physics literature, the resulting…

This paper treats the problem of determining conditions for the Fourier coefficients of a Maass-Hecke newform at cusps other than infinity to be multiplicative. To be precise, the Fourier coefficients are defined using a choice of matrix in…

In the first half of the paper, we lay down a classical approach to the study of Saito-Kurokawa (SK) lifts of (Hecke congruence) square-free level, including the allied new-oldform theory. Our treatment of this relies on a novel idea of…

We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$…

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical…

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for…

This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the…

It has been found experimentally by Brown and Schnetz that the number of points over ${\mathbb F}_p$ of a graph hypersurface is often related to the coefficients of a modular form. In this paper I prove this relation for one example of a…

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues…

We propose an extension of ergodic theory which focuses on the identification of ergodicity in terms of the uniqueness of the invariant measure. We first explain the concept for the doubling maps, which can be analyzed using Fourier…

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism…

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the…

In this paper we give a proof of the {\it Hecke quantum unique ergodicity conjecture} for the multidimensional Berry-Hannay model. A model of quantum mechanics on the 2n-dimensional torus. This result generalizes the proof of the {\it…