Quadratic Forms and Closed Geodesics

Quadratic Forms 

and Closed 

Geodesics 

Y. Petridis 

Outline 

Quadratic Forms and Closed 


Yiannis Petridis 1 

1 University College London 

January 8, 2010

Outline 




Y. Petridis 



Hyperbolic surfaces 

Hyperbolic 

surfaces 

Closed 


Spectral Theory 

Closed Geodesics 

Spectral Theory

Pell’s Equation 




Y. Petridis 

◮ Algorithm for solving 

x 2 − 2y 2 = ±1 

(Pythagoreans) 

◮ Start with 

◮ We get 

x 1 = 1, y 1 = 1 

(x 2 , y 2 ) = (3, 2) 

(x 3 , y 3 ) = (7, 5) 

(x 4 , y 4 ) = (17, 12) 

◮ Recurrences 

x n+1 = x n + 2y n 

y n+1 = x n + y n 

◮ Fundamental solution: 

1 + √ 2 

x n + √ 2y n = (1 + √ 2) n 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 


x 2 − 2y 2 = ±1 



◮ We get 

x 1 = 1, y 1 = 1 

(x 2 , y 2 ) = (3, 2) 

(x 3 , y 3 ) = (7, 5) 

(x 4 , y 4 ) = (17, 12) 


x n+1 = x n + 2y n 

y n+1 = x n + y n 


1 + √ 2 

x n + √ 2y n = (1 + √ 2) n 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 


x 2 − 2y 2 = ±1 



◮ We get 

x 1 = 1, y 1 = 1 

(x 2 , y 2 ) = (3, 2) 

(x 3 , y 3 ) = (7, 5) 

(x 4 , y 4 ) = (17, 12) 


x n+1 = x n + 2y n 

y n+1 = x n + y n 


1 + √ 2 

x n + √ 2y n = (1 + √ 2) n 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 


x 2 − 2y 2 = ±1 



◮ We get 

x 1 = 1, y 1 = 1 

(x 2 , y 2 ) = (3, 2) 

(x 3 , y 3 ) = (7, 5) 

(x 4 , y 4 ) = (17, 12) 


x n+1 = x n + 2y n 

y n+1 = x n + y n 


1 + √ 2 

x n + √ 2y n = (1 + √ 2) n 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 


x 2 − 2y 2 = ±1 



◮ We get 

x 1 = 1, y 1 = 1 

(x 2 , y 2 ) = (3, 2) 

(x 3 , y 3 ) = (7, 5) 

(x 4 , y 4 ) = (17, 12) 


x n+1 = x n + 2y n 

y n+1 = x n + y n 


1 + √ 2 

x n + √ 2y n = (1 + √ 2) n 


Hyperbolic 

surfaces 



Spectral Theory

Indefinite Quadratic Forms 

◮ Q(x, y) = ax 2 + bxy + cy 2 , d = b 2 − 4ac > 0 

( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 

◮ Which integers are represented by Q, i.e. 

Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 




Equivalence of Quadratic Forms 

Q ∼ Q ′ ⇒ they represent the same integers. 

( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 

Q(x + y, y) = Q ′ (x, y) and Q ′ (x − y, y) = Q(x, y)



( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 


Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 






( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 




( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 


Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 






( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 




( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 


Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 






( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 




( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 


Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 






( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 




( ) t ( ) ( ) 

x x a b/2 

◮ Q(x, y) = M M = 

y y 

b/2 c 


Q(x, y) = N, 

x, y, N ∈ N 




Y. Petridis 


Hyperbolic 

surfaces 






( ) 0 −1 

◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S = 

1 0 

◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1 

T = 

0 1 

◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2 


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 



Y. Petridis 


Hyperbolic 

surfaces 



Spectral Theory

Class Numbers 


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 ≠ □} 


ɛ d = t + u√ d 

2 


∑ 

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. 




Y. Petridis 


Hyperbolic 

surfaces 



Spectral Theory

Class Numbers 


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 ≠ □} 


ɛ d = t + u√ d 

2 


∑ 

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. 




Y. Petridis 


Hyperbolic 

surfaces 



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 




Y. Petridis 


Hyperbolic 

surfaces 




* 

(!b!d^(1/2))/2a 

x 

(!b+d^(1/2))/2a

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 




Y. Petridis 


Hyperbolic 

surfaces 




(!b!d^(1/2))/2a 

x 

(!b+d^(1/2))/2a

Linear Fractional Transformations and 

SL 2 (R) 

T (z) = az + b , ad − bc ≠ 0 

cz + d 

maps ( circles) 

to circles 

a b 

γ = 

→ T (z) 

c d 

{ ( a b 

SL 2 (R) = γ = 

c d 

acts on 

Upper-half space 

) 

} 

, ad − bc = 1 

H = {z = x + iy, y > 0} ds 2 = dx 2 + dy 2 

z(t) = x(t) + iy(t), t ∈ [a, b], L = ∫ b 

a 

◮ z → z + 1 

y 

√ 2 

x ′ (t) 2 +y ′ (t) 2 

y(t) 

dt 




Y. Petridis 


Hyperbolic 

surfaces 




◮ z → − 1 z 

Γ = SL 2 (Z) = {γ ∈ SL 2 (R), a, b, c, d ∈ Z}

The Hyperbolic Disc 




Y. Petridis 


◮ Model of hyperbolic 

geometry 

H = {z = x + iy, |z| < 1} 

Hyperbolic 

surfaces 




◮ Hyperbolic metric 

ds 2 = 

dx 2 + dy 2 

(1 − (x 2 + y 2 )) 2 

d(0, z) = ln 1 + |z| 

1 − |z| ,

Geodesics in Hyperbolic Disc 




Y. Petridis 


Hyperbolic 

surfaces 




◮ Semicircles perpendicular to 

boundary 

◮ Diameters

Geodesics in the Upper Half Plane 




Y. Petridis 


Hyperbolic 

surfaces 




(!b!d^(1/2))/2a 

x 

(!b+d^(1/2))/2a

Fundamental Domains 

z ∼ w ⇔ w = T (z), T ∈ Γ 

H/Γ the modular surface 




Y. Petridis 


Hyperbolic 

surfaces 



Spectral Theory

Arithmetic subgroups of SL 2 (Z) 

Example 

Fundamental Domain for Γ 0 (6) 




Y. Petridis 


Hyperbolic 

surfaces 




Hecke subgroups Γ 0 (N) 

az + b 

cz + d ∈ SL 2(Z), N|c

¡ 

Tesselations 




Y. Petridis 


Hyperbolic 

surfaces 

T 1 F 

F 

TF 




T 1 JF 

JF 

T JF 

T 2 UTF 

T 1 U 2 F 

T 1 UTF 

U 2 F 

UTF 

TU 2 F 

1 

0 1 

1 

Figure: Translates of the 

fundamental domain of 

SL 2 (Z) 

Figure: Triangles in the 

disc

Equivalence of Quadratic forms and 

SL 2 (R) 

Q ′ ∼ Q ⇔ M ′ = γ t Mγ, γ ∈ Γ. 

Q → g = 

( t−bu 

) 

2 

−cu 

∈ Γ 

au 

t+bu 

2 

where t 2 − du 2 = 4 and (t, u) is the fundamental 

(smallest solution). 

Remarks: 1. g has eigenvalue ɛ d = t + u√ d 

2 

2. Most g ∈ SL 2 (R) can be diagonalised 

g ∼ 

( N(g) 

1/2 

0 

0 N(g) −1/2 ) 

. 




Y. Petridis 


Hyperbolic 

surfaces 



Spectral Theory

Pell’s Equation and Lengths of Closed 





Y. Petridis 


Hyperbolic 

surfaces 

N(g) 

1 

∫ N(g) 

1 

1 

y dy = ln N(g) = ln(ɛ2 d ) 




x 

1 

N(g) 

Theorem 

The lengths of the closed geodesics for the hyperbolic 

surface H/SL 2 (Z) are 2 log ɛ d with multiplicity h(d), 

d ∈ D.

Distribution Closed geodesics of H/Γ 




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 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 




π(x) ∼ 

x 

◮ π(x) = {p prime, p ≤ x} 

ln x , x → ∞ π(x) ∼ 

x 

ln x , x → ∞ 


∑ 


h(d) ∼ x 2 

2 ln x 


Hyperbolic 

surfaces 



Spectral Theory





Y. Petridis 




π(x) ∼ 

x 

◮ π(x) = {p prime, p ≤ x} 

ln x , x → ∞ π(x) ∼ 

x 

ln x , x → ∞ 


∑ 


h(d) ∼ x 2 

2 ln x 


Hyperbolic 

surfaces 



Spectral Theory

The Laplace Operator 




Y. Petridis 


( ) 

∂ 

∆ = −y 2 2 

∂x 2 + ∂2 

∂y 2 

∆f = 0 ⇔ f is harmonic 

Eigenvalue problem: Solve 

Hyperbolic 

surfaces 




∆f = λf 

Infinite Matrix, no determinant to compute eigenvalues. 

I require f (γz) = f (z), γ ∈ Γ (automorphic form)

Contour plots of eigenfunctions of H/Γ 0 (7) 




Y. Petridis 


Hyperbolic 

surfaces 




Figure: λ = 37.08033 λ = 692.7292

Contour plots of eigenfunctions of H/Γ 0 (3) 




Y. Petridis 


Hyperbolic 

surfaces 




Figure: λ = 26.3467 λ = 60.4397

The Riemann Hypothesis 

◮ Think of 

∞∑ 

n=0 

z n = 1 

1 − z . 

Series: |z| < 1, 1/(1 − z), z ≠ 1. 

∞∑ 1 

◮ ζ(s) = , 

ns R(s) > 1. 

n=1 

Extend to 1/2 ≤ R(s), then to C using functional 

equation 




Y. Petridis 


Hyperbolic 

surfaces 




−1+s Γ((1 − s)/2) 

ζ(s) = π ζ(1 − s). 

Γ(s/2) 

Zeros: ρ, 1 − ρ symmetric around R(s) = 1/2. 

Riemann Hypothesis 

R(ρ) = 1/2 ALWAYS


◮ Think of 

∞∑ 

n=0 

z n = 1 

1 − z . 

Series: |z| < 1, 1/(1 − z), z ≠ 1. 

∞∑ 1 

◮ ζ(s) = , 

ns R(s) > 1. 

n=1 


equation 




Y. Petridis 


Hyperbolic 

surfaces 




−1+s Γ((1 − s)/2) 

ζ(s) = π ζ(1 − s). 

Γ(s/2) 



R(ρ) = 1/2 ALWAYS


◮ Think of 

∞∑ 

n=0 

z n = 1 

1 − z . 

Series: |z| < 1, 1/(1 − z), z ≠ 1. 

∞∑ 1 

◮ ζ(s) = , 

ns R(s) > 1. 

n=1 


equation 




Y. Petridis 


Hyperbolic 

surfaces 




−1+s Γ((1 − s)/2) 

ζ(s) = π ζ(1 − s). 

Γ(s/2) 



R(ρ) = 1/2 ALWAYS

Relations of RH with spectral theory 

Scattering of waves on H/Γ 




Y. Petridis 


Hyperbolic 

surfaces 




φ(s) = √ π 

Γ(s − 1/2)ζ(2s − 1) 

Γ(s)ζ(2s) 

Poles of φ(s) at ρ/2. 

RH⇔Poles of φ(s), have real part 1/4.

Duality between periods and eigenvalues 

Physical background Hyperbolic manifolds Eigenfunctions Periodic orbits Free groups 





Y. Petridis 


Hyperbolic 

surfaces 




Periods 

Eigenvalues







Y. Petridis 


Hyperbolic 

surfaces 




Periods 

Trace Formulae 

Eigenvalues







Y. Petridis 


Hyperbolic 

surfaces 




Periods 

Trace Formulae 

Eigenvalues 

Lengths of closed 

geodesics 

Selberg Trace formula 

Laplace eigenvalues

Simple trace formula 




Y. Petridis 


Hyperbolic 

surfaces 



n∑ 

Tr(A) = a ii = 

n∑ 

i=1 j=1 

What is and how do I compute the trace of an operator? 

λ j 

Spectral Theory

Credits for the pictures 

1. V. Golovshanski, N. Motrov: preprint, Inst. Appl. 

Math. Khabarovsk (1982) 

2. D. Hejhal, B. Rackner: On the topography of Maass 

waveforms for PSL(2, Z ). Experiment. Math. 1 

(1992), no. 4, 275–305. 

3. A. Krieg: http://www.matha.rwthaachen.de/forschung/fundamentalbereich.html 

4. http://mathworld.wolfram.com/ 

5. F. Stromberg: 

http://www.math.uu.se/ fredrik/research/gallery/ 

6. H. Verrill: http://www.math.lsu.edu/ verrill/ 




Y. Petridis 


Hyperbolic 

surfaces 



Spectral Theory

Quadratic Forms and Closed Geodesics

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