## Arithmetic Quantum Chaos: First Annual R.A. Blyth Lectures : March 15-19, 1993 |

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analytic arithmetic groups arithmetic hyperbolic manifolds arithmetic manifolds arithmetic surfaces Assume B-C property behave like random behavior billiard boundary chaotic system classical closed geodesic COM(r compact computed continuous spectrum convexity bound correspondence cosh cusp forms deduce defined density eigenfunctions eigenstates eigenvalues Eisenstein series Euler product example Figure finite functional equation fundamental domain Gaussian geodesic flow Hamiltonian Hecke eigenforms Hecke operators Hejhal hyperbolic surfaces integrable Iwaniec L-functions lattice lectures left hand side Lemma level spacing Lindelof-Hypothesis lower bound Math measure nonarithmetic nonrigid number theory number variance numerical experiments Poisson problems of Q.C. product of degree proof of Theorem Proposition 3.1 quadratic quantum limit quaternion algebra Ramanujan Conjectures random wave model Re(s Riemann zeta function Sarnak satisfies Sato-Tate conjectures Selberg Trace Formula shown singular support SLj(Z spectral stadium Steil strong scarring subgroup Theorem 3.2 theta lifts unique ergodicity unstable periodic orbits zero