Download The Spectrum of Hyperbolic Surfaces by Nicolas Bergeron PDF
By Nicolas Bergeron
This textual content is an advent to the spectral conception of the Laplacian on compact or finite region hyperbolic surfaces. For a few of these surfaces, referred to as “arithmetic hyperbolic surfaces”, the eigenfunctions are of mathematics nature, and one might use analytic instruments in addition to strong tools in quantity conception to review them.
After an advent to the hyperbolic geometry of surfaces, with a distinct emphasis on these of mathematics kind, after which an creation to spectral analytic tools at the Laplace operator on those surfaces, the writer develops the analogy among geometry (closed geodesics) and mathematics (prime numbers) in proving the Selberg hint formulation. in addition to very important quantity theoretic functions, the writer shows purposes of those instruments to the spectral facts of the Laplacian and the quantum targeted ergodicity estate. The latter refers back to the mathematics quantum distinct ergodicity theorem, lately proved through Elon Lindenstrauss.
The fruit of numerous graduate point classes at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces permits the reader to check an array of classical effects after which to be led in the direction of very lively components in sleek mathematics.
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Additional resources for The Spectrum of Hyperbolic Surfaces
Example text
2; Z=pk Z/. This group is made up of matrices with coefficients in Z=pk Z, such that the column vectors are linearly independent. There are p2k 1 choices for the first column vector and, once the first has been fixed, there are p2k pk remaining choices for the second (we exclude the pk multiples of the first vector). p2k pk /. 1 pk 2 /. 2; Z=pskk Z/ ! Z=pskk Z/ . 2; Z=pskk =pskk Z/ is then k equal to p3s pk 2 /. N/ if and only if N D 2. N/ has no elliptic elements. 2; Z/ in the fundamental hyperbolic triangle D are the two vertices of angle =3 and the point i.
For an introduction to these notions the reader can refer to [92, 121] and Serre’s Course in Arithmetic [118], respectively. 2; ZŒ 2/ is not a Fuchsian group. 15 1. 2; Z/nH. ) 2. 1, deduce from the preceding question that a subset of volume 1 lattices in R2 is relatively compact if and only if the height is uniformly bounded from below on this set by a positive constant. 1 and the orbit Ss;t1 i of the point i 2 H under the action of a Siegel set. 1/ Ss;t1 i as soon as p s > 2= 3, t > 1=2. m; n/ to the “Pell-Fermat” Diophantine equation m2 where a is an integer such that an2 D 1; p a … N.
N n X 2/ ˛i : iD1 1. Show that A is invariant under isometries and that if we cut up P into two polygons P1 and P2 along a geodesic, we have A. P/ D A. P1 / C A. P2 /: 2. Show that A. P/ goes to 0 as the diameter of P goes to 0. 3. By mimicking the construction of the Lebesgue measure in the plane, show that A defines a measure in the hyperbolic plane which is invariant under isometries. 4. From the uniqueness of such a measure show that there is a constant c > 0 such that  area. 2; R/ is discrete if and only if n !