Quadratic Forms and Closed Geodesics

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Distribution Closed geodesics of H/Γ Quadratic Forms and Closed Geodesics Y. Petridis Closed geodesics γ. Prime Geodesic Theorem Prime Number Theorem ◮ π(x) = {γ, length (γ) ≤ ln x} π(x) ∼ x ◮ π(x) = {p prime, p ≤ x} ln x , x → ∞ π(x) ∼ x ln x , x → ∞ ◮ Class number distribution (Sarnak, 1982) ∑ d∈D,ɛ d ≤x h(d) ∼ x 2 2 ln x Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory
The Laplace Operator Quadratic Forms and Closed Geodesics Y. Petridis Quadratic Forms ( ) ∂ ∆ = −y 2 2 ∂x 2 + ∂2 ∂y 2 ∆f = 0 ⇔ f is harmonic Eigenvalue problem: Solve Hyperbolic surfaces Closed Geodesics Spectral Theory ∆f = λf Infinite Matrix, no determinant to compute eigenvalues. I require f (γz) = f (z), γ ∈ Γ (automorphic form)
Page 1 and 2: Quadratic Forms and Closed Geodesic
Page 3 and 4: Pell’s Equation Quadratic Forms a
Page 9 and 10: Indefinite Quadratic Forms ◮ Q(x,
Page 15 and 16: Class Numbers h(d) = number of ineq
Page 17 and 18: From Quadratic Forms to Hyperbolic
Page 19 and 20: Linear Fractional Transformations a
Page 21 and 22: Geodesics in Hyperbolic Disc Quadra
Page 23 and 24: Fundamental Domains z ∼ w ⇔ w =
Page 25 and 26: ¡ Tesselations Quadratic Forms and
Page 27 and 28: Pell’s Equation and Lengths of Cl
Page 29: Distribution Closed geodesics of H/
Page 34 and 35: Contour plots of eigenfunctions of
Page 36 and 37: The Riemann Hypothesis ◮ Think of
Page 38 and 39: Relations of RH with spectral theor
Page 40 and 41: Duality between periods and eigenva
Page 42 and 43: Simple trace formula Quadratic Form

Quadratic Forms and Closed Geodesics

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