Quadratic Forms and Closed Geodesics

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Class Numbers h(d) = number of inequivalent forms of discriminant d. h(d) d= 1 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53 2 40, 60, 65, 85, 104, 105, 120, 136, 140, 156, 165 3 229, 257, 316, 321, 469, 473, 568, 733, 761, 892 4 145, 328, 445, 505, 520, 680, 689, 777, 780, 793 5 401, 817, 1093, 1393, 1429, 1641, 1756, 1897, ... ◮ D = {d|d > 0, d ≡ 0, 1 (mod 4), d ≠ □} ◮ x 2 − dy 2 = 4, fundamental solution (t, u) ɛ d = t + u√ d 2 ◮ Gauss, Siegel (1944) ∑ h(d) log ɛ d = π2 x 3/2 + O(x log x). 18ζ(3) d∈D,d≤x ζ(s) = ∞∑ n=1 1 , R(s) > 1. ns Quadratic Forms and Closed Geodesics Y. Petridis Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory
From Quadratic Forms to Hyperbolic Geometry Q(x, y) = ax 2 + bxy + cy 2 az 2 + bz + c = 0 ⇔ z 1 = −b+√ d 2a , z 2 = −b−√ d 2a Quadratic Forms and Closed Geodesics Y. Petridis Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory * (!b!d^(1/2))/2a x (!b+d^(1/2))/2a
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: Class Numbers h(d) = number of ineq
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 and 30: Distribution Closed geodesics of H/
Page 31: The Laplace Operator Quadratic Form
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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