A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical...

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university of copenhagen department of mathematical sciences 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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Page 1: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

Faculty of Science

A Distribution Result Related to AutomorphicForms

Flemming von EssenDepartment of Mathematical Sciences

April 2013

Slide 1/15

Page 2: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

Fuchsian groups

For γ =

(a bc d

)let

γz =az + b

cz + d.

We will consider Fuchsian groups, i.e. discrete subgroups ofSL2(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 upperhalf plane H).

• parabolic if |Trγ| = 2 (or if γ fixes one point inR ∪ {∞}).

• hyperbolic if |Trγ| > 2 (or if γ fixes two points inR ∪ {∞}).

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

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Fuchsian groups

For γ =

(a bc d

)let

γz =az + b

cz + d.

We will consider Fuchsian groups, i.e. discrete subgroups ofSL2(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 upperhalf plane H).

• parabolic if |Trγ| = 2 (or if γ fixes one point inR ∪ {∞}).

• hyperbolic if |Trγ| > 2 (or if γ fixes two points inR ∪ {∞}).

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

Slide 2/15

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un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

Fuchsian groups

For γ =

(a bc d

)let

γz =az + b

cz + d.

We will consider Fuchsian groups, i.e. discrete subgroups ofSL2(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 upperhalf plane H).

• parabolic if |Trγ| = 2 (or if γ fixes one point inR ∪ {∞}).

• hyperbolic if |Trγ| > 2 (or if γ fixes two points inR ∪ {∞}).

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

Slide 2/15

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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 00 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 aRiemann surface, and there is a bijection between conjugacyclasses {γ} = {τγτ−1 | τ ∈ Γ} of hyperbolic elements in Γand closed geodesics on Γ\H.The geodesic associated with γ has length l(γ).

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

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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 00 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 aRiemann surface, and there is a bijection between conjugacyclasses {γ} = {τγτ−1 | τ ∈ Γ} of hyperbolic elements in Γand closed geodesics on Γ\H.

The geodesic associated with γ has length l(γ).

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

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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 00 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 aRiemann surface, and there is a bijection between conjugacyclasses {γ} = {τγτ−1 | τ ∈ Γ} of hyperbolic elements in Γand closed geodesics on Γ\H.The geodesic associated with γ has length l(γ).

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

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), andf : H = {z ∈ C | =z > 0} → C be holomorphic with

f (γz) = f

(az + b

cz + d

)=

ν(γ)

(cz + d)k f (z),

for γ =

(a bc d

)∈ Γ and some k ∈ R. We say that f is

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

withmultiplier system ν. If ν is a multiplier system on Γ thefollowing 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).

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

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), andf : H = {z ∈ C | =z > 0} → C be holomorphic with

f (γz) = f

(az + b

cz + d

)= ν(γ)(cz + d)k f (z),

for γ =

(a bc d

)∈ Γ and some k ∈ R. We say that f is

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

If ν is a multiplier system on Γ thefollowing 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).

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

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Automorphic forms

Let Γ be a discrete subgroup of SL2(R), andf : H = {z ∈ C | =z > 0} → C be holomorphic with

f (γz) = f

(az + b

cz + d

)= ν(γ)(cz + d)k f (z),

for γ =

(a bc d

)∈ Γ and some k ∈ R. We say that f is

an (classical) automorphic form wrt. Γ of weight k withmultiplier system ν. If ν is a multiplier system on Γ thefollowing 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).

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

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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)k f (z),

for γ =

(a bc 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 exponentialgives us a m’th power of f , which is an automorphic form ofweight km, and multiplier system exp(2πikmΦ(γ)).

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

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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)k f (z),

for γ =

(a bc 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 exponentialgives us a m’th power of f , which is an automorphic form ofweight km, and multiplier system exp(2πikmΦ(γ)).

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

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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)k f (z),

for γ =

(a bc 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 exponentialgives us a m’th power of f , which is an automorphic form ofweight km, and multiplier system exp(2πikmΦ(γ)).

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

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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 onSL2(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.

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

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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 onSL2(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.

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

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Ghys’ theoremΓ\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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween γ′ and τ .

(log η)(az+bcz+d

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

The number of times the curves wind around each other.

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Ghys’ theoremΓ\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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween γ′ and τ .

(log η)(az+bcz+d

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

The number of times the curves wind around each other.

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

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Ghys’ theoremΓ\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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween γ′ and τ .

(log η)(az+bcz+d

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

The number of times the curves wind around each other.Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013

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Distribution of the geodesics

P. Sarnak and C. J. Mozzochi has showed that if Φ is as onthe previous slides. So Φ is related to SL2(Z) and η.

Then

limT→∞

#{{γ} | l(γ) ≤ T , a ≤ Φ(γ)/l(γ) ≤ b}#{{γ} | l(γ) ≤ T}

=

arctan(4πb)− arctan(4πa)

π

where for |Trγ| > 2, γ 6= τn and

{γ} = {τγτ−1 | τ ∈ SL2(Z)}.

Can this be generalized to other groups?

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

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Distribution of the geodesics

P. Sarnak and C. J. Mozzochi has showed that if Φ is as onthe previous slides. So Φ is related to SL2(Z) and η.

Then

limT→∞

#{{γ} | l(γ) ≤ T , a ≤ Φ(γ)/l(γ) ≤ b}#{{γ} | l(γ) ≤ T}

=

arctan(4πb)− arctan(4πa)

π

where for |Trγ| > 2, γ 6= τn and

{γ} = {τγτ−1 | τ ∈ SL2(Z)}.

Can this be generalized to other groups?Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013

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Distribution of the geodesics

P. Sarnak and C. J. Mozzochi has showed that if Φ is as onthe previous slides. So Φ is related to SL2(Z) and η.

Then

limT→∞

#{{γ} | l(γ) ≤ T , a ≤ Φ(γ)/l(γ) ≤ b}#{{γ} | l(γ) ≤ T}

=

arctan(4πb)− arctan(4πa)

π

where for |Trγ| > 2, γ 6= τn 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

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Distribution of the geodesicsLet f be a zero free automorphic form on a Fuchsian groupΓ, 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

limT→∞

#{{γ} | l(γ) ≤ T , a ≤ Φ(γ)/l(γ) ≤ b}#{{γ} | l(γ) ≤ T}

=

arctan(4πb)− arctan(4πa)

π

where for |Trγ| > 2, γ 6= τn and

{γ} = {τγτ−1 | τ ∈ Γ}.

Can this be generalized to other groups?

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

Slide 8/15

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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(0 −11 0

), and

(1√

n0 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) = 2η(4z)2

η(2z) areautomorphic forms wrt.

Γ0(4) =

{γ =

(a bc d

) ∣∣∣∣ γ ∈ SL2(Z), 4 | c

}.

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

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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(0 −11 0

), and

(1√

n0 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) = 2η(4z)2

η(2z) areautomorphic forms wrt.

Γ0(4) =

{γ =

(a bc d

) ∣∣∣∣ γ ∈ SL2(Z), 4 | c

}.

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

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Distribution of the geodesicsWe want to prove that.Let f be a zero free automorphic form on a Fuchsian groupΓ, 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

limT→∞

#{{γ} | l(γ) ≤ T , a ≤ Φ(γ)/l(γ) ≤ b}#{{γ} | l(γ) ≤ T}

=

arctan(4πb)− arctan(4πa)

π

where for |Trγ| > 2, γ 6= τn and

{γ} = {τγτ−1 | τ ∈ Γ}.

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

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Idea of the proof - The operator ∆k

If f is a classical automorphic form of weight k, we cancreate another type of automorphic form f ∗ given byf ∗(z) = f (z)(=z)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 = y 2

(∂2

∂x2+

∂2

∂y 2

)− iky

∂x.

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

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Idea of the proof - The operator ∆k

If f is a classical automorphic form of weight k, we cancreate another type of automorphic form f ∗ given byf ∗(z) = f (z)(=z)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 = y 2

(∂2

∂x2+

∂2

∂y 2

)− iky

∂x.

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

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, andh 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)−1/2g(l(γ))

+some other terms.

Here the r 2n + 1/4 is the eigenvalues of ∆k .

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

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

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, andh 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)−1/2g(l(γ))

+some other terms.

Here the r 2n + 1/4 is the eigenvalues of ∆k .

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

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Selberg’s trace formula

Theorem (Selberg’s trace formula)

If g a smooth function that decreases sufficiently quick, andh 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)−1/2g(l(γ))

+some other terms.

Here the r 2n + 1/4 is the eigenvalues of ∆k .

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

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

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The eigenvaluesWe want to estimate the contribution from the eigenvalues

∞∑n=0

h(rn).

• It turns out that we only need to consider smalleigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes fromour 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 eigenvaluehas multiplicity 1 and the next eigenvalue is boundedfrom below.

• For weight 0, the multiplicity is 1. So it is enough toshow that, the eigenvalues are continuous in 0.

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

Slide 13/15

Page 32: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

The eigenvaluesWe want to estimate the contribution from the eigenvalues

∞∑n=0

h(rn).

• It turns out that we only need to consider smalleigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes fromour 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 eigenvaluehas multiplicity 1 and the next eigenvalue is boundedfrom below.

• For weight 0, the multiplicity is 1. So it is enough toshow that, the eigenvalues are continuous in 0.

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

Slide 13/15

Page 33: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

The eigenvaluesWe want to estimate the contribution from the eigenvalues

∞∑n=0

h(rn).

• It turns out that we only need to consider smalleigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes fromour 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 eigenvaluehas multiplicity 1 and the next eigenvalue is boundedfrom below.

• For weight 0, the multiplicity is 1. So it is enough toshow that, the eigenvalues are continuous in 0.

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

Slide 13/15

Page 34: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

The eigenvaluesWe want to estimate the contribution from the eigenvalues

∞∑n=0

h(rn).

• It turns out that we only need to consider smalleigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes fromour 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 eigenvaluehas multiplicity 1 and the next eigenvalue is boundedfrom below.

• For weight 0, the multiplicity is 1. So it is enough toshow that, the eigenvalues are continuous in 0.

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

Slide 13/15

Page 35: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

The eigenvaluesWe want to estimate the contribution from the eigenvalues

∞∑n=0

h(rn).

• It turns out that we only need to consider smalleigenvalues, since h is decreasing rapidly.

• For small weight m, the smallest eigenvalue comes fromour 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 eigenvaluehas multiplicity 1 and the next eigenvalue is boundedfrom below.

• For weight 0, the multiplicity is 1. So it is enough toshow that, the eigenvalues are continuous in 0.

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

Slide 13/15

Page 36: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

End of the sketched proofThe lower bound on the eigenvalues, enables us to estimatethe

∑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 withexp(2πikn) and integrate, we can estimate∑

{γ}N(γ)≤xΦ(γ)=n

l(γ) = cΓ

∫ x

2

ln y

(4πn/N)2 + (ln y)2dy + O(x1−δΓ).

Working with this we get our distribution result.

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

Slide 14/15

Page 37: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

End of the sketched proofThe lower bound on the eigenvalues, enables us to estimatethe

∑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 withexp(2πikn) and integrate, we can estimate∑

{γ}N(γ)≤xΦ(γ)=n

l(γ) = cΓ

∫ x

2

ln y

(4πn/N)2 + (ln y)2dy + O(x1−δΓ).

Working with this we get our distribution result.

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

Slide 14/15

Page 38: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

End of the sketched proofThe lower bound on the eigenvalues, enables us to estimatethe

∑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 withexp(2πikn) and integrate, we can estimate∑

{γ}N(γ)≤xΦ(γ)=n

l(γ) = cΓ

∫ x

2

ln y

(4πn/N)2 + (ln y)2dy + O(x1−δΓ).

Working with this we get our distribution result.Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013

Slide 14/15

Page 39: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

InterpretationsWe proved

limT→∞

#{{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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween 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

Page 40: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

InterpretationsWe proved

limT→∞

#{{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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween 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

Page 41: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

InterpretationsWe proved

limT→∞

#{{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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween 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

Page 42: A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical sciences Faculty of Science A Distribution Result Related to Automorphic Forms

un i ver s i ty of copenhagen department of mathemat i ca l sc i ence s

InterpretationsWe proved

limT→∞

#{{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′ inS3\τ associated with γ, and 12Φ(γ) is the linking numberbetween 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

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