A Distribution Result Related to Automorphic Formsuniversity of copenhagendepartment of mathematical...
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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
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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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
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
Slide 3/15
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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
Slide 3/15
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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
Slide 3/15
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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
Slide 4/15
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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
Slide 4/15
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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
Slide 4/15
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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
Slide 5/15
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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
Slide 5/15
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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
Slide 5/15
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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
Slide 6/15
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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.
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
Slide 7/15
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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
Slide 7/15
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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
Slide 8/15
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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
Slide 8/15
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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
Slide 8/15
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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
Slide 9/15
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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
Slide 9/15
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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
Slide 10/15
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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
Slide 11/15
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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
Slide 12/15
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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
Slide 12/15
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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
Slide 12/15
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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
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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
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
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
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
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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
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
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
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
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
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
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