By Nicolas Bergeron
This textual content is an advent to the spectral idea of the Laplacian on compact or finite zone hyperbolic surfaces. For a few of these surfaces, referred to as “arithmetic hyperbolic surfaces”, the eigenfunctions are of mathematics nature, and one may possibly use analytic instruments in addition to robust equipment in quantity concept to check them.
After an advent to the hyperbolic geometry of surfaces, with a distinct emphasis on these of mathematics kind, after which an advent to spectral analytic equipment 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. besides very important quantity theoretic functions, the writer indicates purposes of those instruments to the spectral records of the Laplacian and the quantum special ergodicity estate. The latter refers back to the mathematics quantum certain ergodicity theorem, lately proved by way of Elon Lindenstrauss.
The fruit of numerous graduate point classes at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces permits the reader to study an array of classical effects after which to be led in the direction of very lively parts in sleek mathematics.
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Additional resources for The Spectrum of Hyperbolic Surfaces
Q; Q/ 38 2 Arithmetic Hyperbolic Surfaces which induces an isomorphism on the real groups1 D1a;b =f˙1g ! q/: Proof It suffices to construct the first isomorphism. Q/ by interior automorphisms. P/ D P. S˛ /jP . Q/ ! Q/ D Q . Before continuing we make a few preliminary calculations. x 2 P/: Already we see that if ˇ is the rotation of angle with axis ˇ, we have ˇ D ˇ D sˇ . q; Q/. ˇi / ¤ 0, the integer r being even if u is in SO and odd otherwise. x/ D . 2; 1/ the connected component of the identity of the special orthogonal group of the quadratic form q on P.
One can then define, by induction on n, the notion of an admissible family of vectors in L. This is a family . f1 ; : : : ; fn / of vectors in L such that • f1 is a vector in L f0g of minimal norm, • the images fP2 ; : : : ; fPn of f2 ; : : : ; fn in the lattice LP WD L=Zf1 of the Euclidean space P and Rn =Rf1 form an admissible family of L, • the vectors fi are of minimal norm among the vectors of L whose image in LP is fPi . ) It is a standard fact that the Z-module L admits an admissible family .
Assume for example that ax21 is non-zero. x21 bx23 /. x21 bx23 /; x0 x1 C bx2 x3 / is an integer solution to ax21 bx22 C abx23 D 0. x1 ; 0; x3 / is a non-zero integer solution (since x1 ¤ 0) to ax21 bx22 C abx23 D 0. 7, the quadratic forms q D ax21 bx22 C abx23 and q0 D x21 x22 C x33 are similar over Q. 3; Q/. q0 ; Z/. 2; R/ then implies that the groups a;b and 1;1 are commensurable in G. 2; Z/. 2; Z/ if and only if ax2 by C abz2 is similar to x2 y2 C z2 . On the other hand 2 ab. abz/2 ; so that the quadratic forms ax2 by2 C abz2 and x2 ay2 bz2 are themselves similar.
The Spectrum of Hyperbolic Surfaces by Nicolas Bergeron