Morten S. Risager

On the distribution of modular symbols for compact surfaces

We prove that the modular symbols appropriately normalized and ordered have an asymptotical normal distribution for all cocompact subgroups of SL2(R). We introduce hyperbolic Eisenstein series in order to calculate the moments of the modular symbols.

Equidistribution of geodesics on homology classes and analogues for free groups

Abstract We investigate how often geodesics have homology in a fixed set of the homology lattice of a compact Riemann surface. We prove that closed geodesics are equidistributed on any set with asymptotic density with respect to a specific norm. We explain the analogues for free groups, conjugacy classes and discrete logarithms, in particular, we investigate the density of conjugacy classes with relatively prime discrete logarithms.

Quantum Limits of Eisenstein Series and Scattering States

We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

The distribution of values of the Poincaré pairing for hyperbolic Riemann surfaces

Abstract For a cocompact group of SL2(ℝ) we fix a non-zero harmonic 1-form α. We normalize and order the values of the Poincaré pairing 〈 γ,α〉 according to the length of the corresponding closed geodesic l(γ). We prove that these normalized values have a Gaussian distribution.

Discrete logarithms in free groups

For the free group on n generators we prove that the discrete logarithm is distributed according to the standard Gaussian when the logarithm is renormalized appropriately.

Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series

The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for Γ0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.

Dissolving of cusp forms: higher-order Fermi's golden rules

For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to become resonances. A sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermis Golden Rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction

Non-vanishing of Taylor coefficients and Poincaré series

u_j

Arithmetic statistics of modular symbols

into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the

ON SELBERG'S SMALL EIGENVALUE CONJECTURE AND RESIDUAL EIGENVALUES

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