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Faculty of Science

A Distribution Result Related to Automorphic Forms

Flemming von Essen

Department of Mathematical Sciences

April 2013 Slide 1/15

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Fuchsian groups

For γ = a b c d

• let

γz = az + b cz + d .

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Fuchsian groups

For γ = a b c d

• let

γz = az + b cz + d . We will consider Fuchsian groups, i.e. discrete subgroups of SL2(R).

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Fuchsian groups

For γ = a b c d

• let

γz = az + b cz + d . We will consider Fuchsian groups, i.e. discrete subgroups of SL2(R). Let Γ be such a group. We say that γ ∈ Γ\{±I} is

• elliptic if |Trγ| < 2 (or if γ fixes a point in the upper

half plane H).

• parabolic if |Trγ| = 2 (or if γ fixes one point in

R ∪ {∞}).

• hyperbolic if |Trγ| > 2 (or if γ fixes two points in

R ∪ {∞}).

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Hyperbolic matrices and geodesics

Let γ be a hyperbolic (i.e. |Trγ| > 2).Then there exists |λ| > 1 and A ∈ SL2(R) s.t. γ = A λ λ−1

• A−1,

and we define N(γ) := λ2 and l(γ) = log N(γ).

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Hyperbolic matrices and geodesics

Let γ be a hyperbolic (i.e. |Trγ| > 2).Then there exists |λ| > 1 and A ∈ SL2(R) s.t. γ = A λ λ−1

• A−1,

and we define N(γ) := λ2 and l(γ) = log N(γ). Let Γ be a discrete subgroup of SL2(R).Then Γ\H is a Riemann surface, and there is a bijection between conjugacy classes {γ} = {τγτ −1 | τ ∈ Γ} of hyperbolic elements in Γ and closed geodesics on Γ\H.

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Hyperbolic matrices and geodesics

Let γ be a hyperbolic (i.e. |Trγ| > 2).Then there exists |λ| > 1 and A ∈ SL2(R) s.t. γ = A λ λ−1

• A−1,

and we define N(γ) := λ2 and l(γ) = log N(γ). Let Γ be a discrete subgroup of SL2(R).Then Γ\H is a Riemann surface, and there is a bijection between conjugacy classes {γ} = {τγτ −1 | τ ∈ Γ} of hyperbolic elements in Γ and closed geodesics on Γ\H. The geodesic associated with γ has length l(γ).

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), and f : H = {z ∈ C | ℑz > 0} → C be holomorphic with f (γz) = f az + b cz + d

• =

(cz + d)kf (z), for γ = a b c d

• ∈ Γ and some k ∈ R. We say that f is

an (classical) automorphic form wrt. Γ of weight k.

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), and f : H = {z ∈ C | ℑz > 0} → C be holomorphic with f (γz) = f az + b cz + d

• = ν(γ)(cz + d)kf (z),

for γ = a b c d

• ∈ Γ and some k ∈ R. We say that f is

an (classical) automorphic form wrt. Γ of weight k with multiplier system ν.

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), and f : H = {z ∈ C | ℑz > 0} → C be holomorphic with f (γz) = f az + b cz + d

• = ν(γ)(cz + d)kf (z),

for γ = a b c d

• ∈ Γ and some k ∈ R. We say that f is

an (classical) automorphic form wrt. Γ of weight k with multiplier system ν. If ν is a multiplier system on Γ the following should hold 1) |ν(γ)| = 1, for all γ ∈ Γ, 2) ν(−I) = exp(−πik) if −I ∈ Γ, 3) ν(γ1γ2) = σk(γ1, γ2)ν(γ1)ν(γ2).

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Zero free automorphic forms

Let f : H → C be a zero free automorphic form, so f az + b cz + d

• = ν(γ)(cz + d)kf (z),

for γ = a b c d

• ∈ Γ.

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Zero free automorphic forms

Let f : H → C be a zero free automorphic form, so f az + b cz + d

• = ν(γ)(cz + d)kf (z),

for γ = a b c d

• ∈ Γ. We can take a holomorphic

logarithm log f az + b cz + d

• = 2πikΦ(γ) + k log(cz + d) + log f (z),

(1) where exp(2πikΦ(γ)) = ν(γ).

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Zero free automorphic forms

Let f : H → C be a zero free automorphic form, so f az + b cz + d

• = ν(γ)(cz + d)kf (z),

for γ = a b c d

• ∈ Γ. We can take a holomorphic

logarithm log f az + b cz + d

• = 2πikΦ(γ) + k log(cz + d) + log f (z),

(1) where exp(2πikΦ(γ)) = ν(γ). Multiplying with m ∈ R in (1) and taking the exponential gives us a m’th power of f , which is an automorphic form of weight km, and multiplier system exp(2πikmΦ(γ)).

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The Dedekind η-function

Let η : H → C be defined by η(z) = exp πiz 12 ∞

• n=1

(1 − exp(2πinz)), then η is a zero free weight 1/2 automorphic form on SL2(Z).

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The Dedekind η-function

Let η : H → C be defined by η(z) = exp πiz 12 ∞

• n=1

(1 − exp(2πinz)), then η is a zero free weight 1/2 automorphic form on SL2(Z).Taking logarithms as on the previous slide we get (log η) az + b cz + d

• = πiΦ(γ) + log(cz + d)/2 + (log η)(z),

where 12Φ is the so called Rademacher function.

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Ghys’ theorem

Γ\H is homeomorphic to {(x, y) ∈ C 2 | |x|2 + |y|2 = 1}\τ, where τ is a trefoil knot.

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Ghys’ theorem

Γ\H is homeomorphic to {(x, y) ∈ C 2 | |x|2 + |y|2 = 1}\τ, where τ is a trefoil knot.

Theorem (´ E Ghys)

For γ ∈ SL2(Z) hyperbolic, there is a (oriented) curve γ′ in S3\τ associated with γ, and 12Φ(γ) is the linking number between γ′ and τ. (log η)

• az+b

cz+d

• = πiΦ(γ) + log(cz + d)/2 + (log η)(z),

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Ghys’ theorem

Γ\H is homeomorphic to {(x, y) ∈ C 2 | |x|2 + |y|2 = 1}\τ, where τ is a trefoil knot.

Theorem (´ E Ghys)

For γ ∈ SL2(Z) hyperbolic, there is a (oriented) curve γ′ in S3\τ associated with γ, and 12Φ(γ) is the linking number between γ′ and τ. (log η)

• az+b

cz+d

• = πiΦ(γ) + log(cz + d)/2 + (log η)(z),

The number of times the curves wind around each other.

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Distribution of the geodesics

• P. Sarnak and C. J. Mozzochi has showed that if Φ is as on

the previous slides. So Φ is related to SL2(Z) and η. Then lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π where for |Trγ| > 2, γ = τ n and {γ} = {τγτ −1 | τ ∈ SL2(Z)}.

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Distribution of the geodesics

• P. Sarnak and C. J. Mozzochi has showed that if Φ is as on

the previous slides. So Φ is related to SL2(Z) and η. Then lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π where for |Trγ| > 2, γ = τ n and {γ} = {τγτ −1 | τ ∈ SL2(Z)}. Can this be generalized to other groups?

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Distribution of the geodesics

• P. Sarnak and C. J. Mozzochi has showed that if Φ is as on

the previous slides. So Φ is related to SL2(Z) and η. Then lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π where for |Trγ| > 2, γ = τ n and {γ} = {τγτ −1 | τ ∈ SL2(Z)}. Can this be generalized to other groups? Yes.

Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 8/15

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Distribution of the geodesics

Let f be a zero free automorphic form on a Fuchsian group Γ, such that µ(Γ\H) < ∞, Φ : Γ → Q and log f az + b cz + d

• = 2πikΦ(γ) + k log(cz + d) + log f (z).

Then lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π where for |Trγ| > 2, γ = τ n and {γ} = {τγτ −1 | τ ∈ Γ}.

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Examples of zero free automorphic forms

Let n = 1, 2, 3, 4, then η

• z

√n

• η(√nz) is a zero free

automorphic form, wrt. the group generated by −1 1

• , and

1 √n 1

• .

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Examples of zero free automorphic forms

Let n = 1, 2, 3, 4, then η

• z

√n

• η(√nz) is a zero free

automorphic form, wrt. the group generated by −1 1

• , and

1 √n 1

• .

θ(z) =

n∈Z exp(2πin2z) = η(2z)5 η(z)2η(4z)2

θM(z) =

n∈Z(−1)n exp(2πin2z) = η(z)2 η(2z)

and θF(z) =

n∈Z exp(2πi(n + 1/2)2z) = 2 η(4z)2 η(2z) are

automorphic forms wrt. Γ0(4) =

• γ =

a b c d

• γ ∈ SL2(Z), 4 | c
• .

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Distribution of the geodesics

We want to prove that. Let f be a zero free automorphic form on a Fuchsian group Γ, such that µ(Γ\H) < ∞, Φ : Γ → Q and log f az + b cz + d

• = 2πikΦ(γ) + k log(cz + d) + log f (z).

Then lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π where for |Trγ| > 2, γ = τ n and {γ} = {τγτ −1 | τ ∈ Γ}.

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Idea of the proof - The operator ∆k

If f is a classical automorphic form of weight k, we can create another type of automorphic form f ∗ given by f ∗(z) = f (z)(ℑz)k/2, which transforms in the following way f ∗ az + b cz + d

• = ν(γ)

cz + d |cz + d| k f ∗(z).

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Idea of the proof - The operator ∆k

If f is a classical automorphic form of weight k, we can create another type of automorphic form f ∗ given by f ∗(z) = f (z)(ℑz)k/2, which transforms in the following way f ∗ az + b cz + d

• = ν(γ)

cz + d |cz + d| k f ∗(z). Then is f ∗ a eigenfunction of ∆k given by ∆k = y2 ∂2 ∂x2 + ∂2 ∂y2

• − iky ∂

∂x .

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, and h is the inverse Fourier transform of g, then

∞

• n=0

h(rn) =

• {γ}

Trγ>2

ν(γ)l(γ0) N(γ)1/2 − N(γ)−1/2 g(l(γ)) +some other terms.

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, and h is the inverse Fourier transform of g, then

∞

• n=0

h(rn) =

• {γ}

Trγ>2

ν(γ)l(γ0) N(γ)1/2 − N(γ)−1/2 g(l(γ)) +some other terms. Here the r2

n + 1/4 is the eigenvalues of ∆k.

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, and h is the inverse Fourier transform of g, then

∞

• n=0

h(rn) =

• {γ}

Trγ>2

ν(γ)l(γ0) N(γ)1/2 − N(γ)−1/2 g(l(γ)) +some other terms. Here the r2

n + 1/4 is the eigenvalues of ∆k.

To prove the theorem we use a family of g’s that is approximately indicator functions, and multiplier systems on the form ν(γ) = exp(2πikΦ(γ)), for arbitrary k. Then we do a lot of estimations on the terms in the formula.

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The eigenvalues

We want to estimate the contribution from the eigenvalues

∞

• n=0

h(rn).

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The eigenvalues

We want to estimate the contribution from the eigenvalues

∞

• n=0

h(rn).

• It turns out that we only need to consider small

eigenvalues, since h is decreasing rapidly.

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The eigenvalues

We want to estimate the contribution from the eigenvalues

∞

• n=0

h(rn).

• It turns out that we only need to consider small

eigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes from
• ur original (f ∗)m/k, where f is our zero free classical

automorphic form. This eigenvalue is |m|

2

• 1 − |m|

2

• .

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The eigenvalues

We want to estimate the contribution from the eigenvalues

∞

• n=0

h(rn).

• It turns out that we only need to consider small

eigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes from
• ur original (f ∗)m/k, where f is our zero free classical

automorphic form. This eigenvalue is |m|

2

• 1 − |m|

2

• .
• We need to show that for small weight, this eigenvalue

has multiplicity 1 and the next eigenvalue is bounded from below.

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The eigenvalues

We want to estimate the contribution from the eigenvalues

∞

• n=0

h(rn).

• It turns out that we only need to consider small

eigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes from
• ur original (f ∗)m/k, where f is our zero free classical

automorphic form. This eigenvalue is |m|

2

• 1 − |m|

2

• .
• We need to show that for small weight, this eigenvalue

has multiplicity 1 and the next eigenvalue is bounded from below.

• For weight 0, the multiplicity is 1. So it is enough to

show that, the eigenvalues are continuous in 0.

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End of the sketched proof

The lower bound on the eigenvalues, enables us to estimate the

n h(rn). This along with estimates on the rest of the

terms in the trace formula and summation by parts gives us

• {γ}

N(γ)≤x

ν(γ)l(γ) = x1−|k|/2 1 − |k|/21[0,kδ](|k|) + O

• x1−δΓ log 1

|k|

• .

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End of the sketched proof

The lower bound on the eigenvalues, enables us to estimate the

n h(rn). This along with estimates on the rest of the

terms in the trace formula and summation by parts gives us

• {γ}

N(γ)≤x

ν(γ)l(γ) = x1−|k|/2 1 − |k|/21[0,kδ](|k|) + O

• x1−δΓ log 1

|k|

• .

Here ν(γ) = exp(2πikΦ(γ)), so if we multiply with exp(2πikn) and integrate, we can estimate

• {γ}

N(γ)≤x Φ(γ)=n

l(γ) = cΓ x

2

ln y (4πn/N)2 + (ln y)2 dy + O(x1−δΓ).

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End of the sketched proof

The lower bound on the eigenvalues, enables us to estimate the

n h(rn). This along with estimates on the rest of the

terms in the trace formula and summation by parts gives us

• {γ}

N(γ)≤x

ν(γ)l(γ) = x1−|k|/2 1 − |k|/21[0,kδ](|k|) + O

• x1−δΓ log 1

|k|

• .

Here ν(γ) = exp(2πikΦ(γ)), so if we multiply with exp(2πikn) and integrate, we can estimate

• {γ}

N(γ)≤x Φ(γ)=n

l(γ) = cΓ x

2

ln y (4πn/N)2 + (ln y)2 dy + O(x1−δΓ). Working with this we get our distribution result.

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Interpretations

We proved lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π .

Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 15/15

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

Interpretations

We proved lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π . For Γ = SL2(Z), this was interesting due to Ghys’ theorem

Theorem (´ E Ghys)

For γ ∈ SL2(Z) hyperbolic, there is a (oriented) curve γ′ in S3\τ associated with γ, and 12Φ(γ) is the linking number between γ′ and τ.

Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 15/15

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

Interpretations

We proved lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π . For Γ = SL2(Z), this was interesting due to Ghys’ theorem

Theorem (´ E Ghys)

For γ ∈ SL2(Z) hyperbolic, there is a (oriented) curve γ′ in S3\τ associated with γ, and 12Φ(γ) is the linking number between γ′ and τ. Is there an interpretation of Φ for other groups?

Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 15/15

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s

Interpretations

We proved lim

T→∞

#{{γ} | l(γ) ≤ T, a ≤ Φ(γ)/l(γ) ≤ b} #{{γ} | l(γ) ≤ T} = arctan(4πb) − arctan(4πa) π . For Γ = SL2(Z), this was interesting due to Ghys’ theorem

Theorem (´ E Ghys)

For γ ∈ SL2(Z) hyperbolic, there is a (oriented) curve γ′ in S3\τ associated with γ, and 12Φ(γ) is the linking number between γ′ and τ. Is there an interpretation of Φ for other groups? Yes (at least for some groups).

Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 15/15