694 research outputs found
On pairs of prime geodesics with fixed homology difference
We exhibit the analogy between prime geodesics on hyperbolic Riemann surfaces
and ordinary primes. We present new asymptotic counting results concerning
pairs of prime geodesics whose homology difference is fixed.Comment: 19 pages, Corrected typos, corrected MSC-clas
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 SL_2(R). We
introduce hyperbolic Eisenstein series in order to calculate the moments of the
modular symbols.Comment: 25 page
Dissolving 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 Fermi's Golden
Rule. We prove formulas for higher approximations and obtain necessary and
sufficient conditions for dissolving a cusp form with eigenfunction into
a resonance. In the framework of perturbations in character varieties, we
relate the result to the special values of the -series . This is the Rankin-Selberg convolution of with , where
is the antiderivative of a weight 2 cusp form. In an example we show
that the above-mentioned conditions force the embedded eigenvalue to become a
resonance in a punctured neighborhood of the deformation space.Comment: 33 pages, typos corrected, new section adde
On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces
We study a variant of a problem considered by Dinaburg and Sinai on the
statistics of the minimal solution to a linear Diophantine equation. We show
that the signed ratio between the Euclidean norms of the minimal solution and
the coefficient vector is uniformly distributed modulo one. We reduce the
problem to an equidistribution theorem of Anton Good concerning the orbits of a
point in the upper half-plane under the action of a Fuchsian group.Comment: Minor changes. Final version to appear in proceedings of the
conference on the occasion of Sunada's 60th birthday. Contemp.Math. series of
Amer. Math.So
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