1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FINITE MODELS 347 following discussion p is always an odd prime to make life simpler. We begin with finite analogues ...

348 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS of the Kloosterman sums becomes semi-circle in the limit for large p. The top part ...

LECTURE 1. FINITE MODELS Spectrum for E48011~,1) 1000 500 0 ·1 .0.8 -06 -04 - 02 0 .2 04 0.6 0.8 Level Spacing for E48011 ~.1) 4 ...

350 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS This is a natural finite analogue of the Poincare metric for the classical hyperbo ...

LECTURE 1. FINITE MODELS spectrum finite upper half plane, p=353, a=3, g=1, h=5 40 20 -30 .L{) ·10 10 20 30 40 level spacing fin ...

352 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS spectrum for finite upper half plane over ring Zl169Z, d=2,a=17 100 80 60 40 20 (^ ...

LECTURE 1. FINITE MODELS 353 consists of matrices 0 1 y , with x, y , z E F. When F is the field of real ( 1 x z ) 0 0 1 numbers ...

354 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS M. Minei [60] has noticed that one can draw very different pictures of the spec- t ...

Lecture 2. Three symmetric spaces Zeta Functions of Graphs First let us consider the graph theoretic analogue of Selberg's zet ...

356 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS Terras [73], [74] for some elementary ones. A survey with lots of references is to ...

LECTURE 2. THREE SYMMETRIC SPACES 357 and regular. Here "GUE" means that the spacing between pairs of zeros/poles is that of the ...

358 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS • • H T Figure 1. The 3 Symmetric Spaces. The Poincare upper half plane H is o ...

LECTURE 2. THREE SYMMETRIC SPACES 359 In Figure 2 we show t he finite upper h alf plane graph for q = 3 (the octahedron) and one ...

360 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS . -. • H • • • • • T Figure 3. Geodesics in the 3 Symmetric Spaces H, T , and ...

LECTURE 2. THREE SYMMETRIC SPACES 361 II Poincare Upper H alf Plane I (q + 1)-regular Tree space= H = space= T = {z = x + iy Ix, ...

362 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS II Finite Upper Half Plane II space= Hq = {z=x+v;5y I x,yElFq,y#O} where 0 # a^2 , ...

LECTURE 2. THREE SYl\IIMETRIC SPACES 363 The spherical transform of a function fin L^2 (K\G/ K); i.e., a K -bi-invariant functio ...

364 AUDREY TERRAS, ARITHiVIETICAL QUANTUM CHAOS r discrete subgroup of G r c G, funda mental group of X with f \ H compact, X = ...

LECTURE 2. THREE SYlVIMETRIC SPACES 365 r c G ; r = GL(2, 1Fp) conjugacy clas s e s in r central { ~ ~ ) , a E lF; hy p erbolic ...

366 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS The \iVeyl law implies that cuspid al Maass wave forms exist when r\H is compact. ...