A bound for the number of automorphisms of an arithmetic...
Transcript of A bound for the number of automorphisms of an arithmetic...
![Page 1: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/1.jpg)
Preliminaries The Lower Bound Effective Version
A bound for the number of automorphisms ofan arithmetic Riemann surface
A paper by Mikhail Belolipetsky and Gareth Jones
David Roe
Harvard University
April 1, 2008
Automorphisms of Arithmetic Riemann Surfaces
![Page 2: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/2.jpg)
Preliminaries The Lower Bound Effective Version
Arizona Winter School Project
Project MembersLinda Gruendken, Guillermo Mantilla, Dermot McCarthy,
David Roe, Kate Stange, Ying Zong, and Maryna Viazovska
Project SupervisorsPaula Tretkoff and Ahmad El-Guindy
Automorphisms of Arithmetic Riemann Surfaces
![Page 3: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/3.jpg)
Preliminaries The Lower Bound Effective Version
Outline
1 Definitions, Geometric Preliminaries and an Example
2 A Sharp Lower Bound on Nar (g)
3 An Effective Version of the Main Theorem
Automorphisms of Arithmetic Riemann Surfaces
![Page 4: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/4.jpg)
Preliminaries The Lower Bound Effective Version
Outline
1 Definitions, Geometric Preliminaries and an Example
2 A Sharp Lower Bound on Nar (g)
3 An Effective Version of the Main Theorem
Automorphisms of Arithmetic Riemann Surfaces
![Page 5: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/5.jpg)
Preliminaries The Lower Bound Effective Version
We will be considering automorphism groups of compactRiemann surfaces of genus g ≥ 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 6: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/6.jpg)
Preliminaries The Lower Bound Effective Version
We will be considering automorphism groups of compactRiemann surfaces of genus g ≥ 2.
Definition
Define N(g) as the supremum of |Aut(S)| among all surfaces Sof genus g.
Automorphisms of Arithmetic Riemann Surfaces
![Page 7: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/7.jpg)
Preliminaries The Lower Bound Effective Version
We will be considering automorphism groups of compactRiemann surfaces of genus g ≥ 2.
Definition
Define N(g) as the supremum of |Aut(S)| among all surfaces Sof genus g.
Theorem (Hurwitz, Accola, Maclachlan)
We have8(g + 1) ≤ N(g) ≤ 84(g − 1),
the upper bound due to Hurwitz, and the lower bound due toAccola and Maclachlan.
Automorphisms of Arithmetic Riemann Surfaces
![Page 8: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/8.jpg)
Preliminaries The Lower Bound Effective Version
By the uniformization theorem, each surface with g ≥ 2 can berepresented as
S = ΓS\H,
where H is the hyperbolic upper half plane and ΓS is acocompact torsion-free discrete subgroup of PSL2(R).
Automorphisms of Arithmetic Riemann Surfaces
![Page 9: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/9.jpg)
Preliminaries The Lower Bound Effective Version
By the uniformization theorem, each surface with g ≥ 2 can berepresented as
S = ΓS\H,
where H is the hyperbolic upper half plane and ΓS is acocompact torsion-free discrete subgroup of PSL2(R).
We will be restricting our attention to the arithmetic surfaces:those coming from arithmetic subgroups ΓS .
Automorphisms of Arithmetic Riemann Surfaces
![Page 10: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/10.jpg)
Preliminaries The Lower Bound Effective Version
Definition
Let K be a totally real number field, let a, b ∈ K , and letA = (a,b
K ) be a quaternion algebra. Suppose that we have
ρ : (a,bR
) → M2(R) an isomorphism and (σ(a),σ(b)R
) ∼= H for everynon-identity σ : K → R. Let O be an order in A and let O1 bethe elements of norm 1 in O. We call a subgroup of PSL(2, R)that is commensurable with the image in PSL(2, R) of someρ(O1) an arithmetic subgroup.
An arithmetic surface is a Riemann surface S that can beexpressed as S ∼= ΓS\H with ΓS arithmetic. A non-arithmeticsurface is one that cannot be expressed in this way.
Nar (g) = sup{|Aut(S)| : S genus g, arithmetic},
Nnar (g) = sup{|Aut(S)| : S genus g, non-arithmetic}.
Automorphisms of Arithmetic Riemann Surfaces
![Page 11: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/11.jpg)
Preliminaries The Lower Bound Effective Version
We can split the bounds on N(g) into bounds for arithmetic andnon-arithmetic surfaces.
Automorphisms of Arithmetic Riemann Surfaces
![Page 12: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/12.jpg)
Preliminaries The Lower Bound Effective Version
We can split the bounds on N(g) into bounds for arithmetic andnon-arithmetic surfaces.
Theorem (Hurwitz, Accola, Maclachlan, Belolipetsky, Jones)
4(g − 1) ≤ Nar (g) ≤ 84(g − 1)
8(g + 1) ≤ Nnar (g) ≤156
7(g − 1)
Automorphisms of Arithmetic Riemann Surfaces
![Page 13: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/13.jpg)
Preliminaries The Lower Bound Effective Version
We can split the bounds on N(g) into bounds for arithmetic andnon-arithmetic surfaces.
Theorem (Hurwitz, Accola, Maclachlan, Belolipetsky, Jones)
4(g − 1) ≤ Nar (g) ≤ 84(g − 1)
8(g + 1) ≤ Nnar (g) ≤156
7(g − 1)
We will be concerned with the lower arithmetic bound.
Automorphisms of Arithmetic Riemann Surfaces
![Page 14: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/14.jpg)
Preliminaries The Lower Bound Effective Version
We call a discrete subgroup of PSL2(R) a Fuchsian group. Anycocompact Fuchsian group has a presentation
Γ(g; m1, . . . , mk ) = 〈α1, β1, . . . , αg , βg , γ1, . . . γk |
g∏
i=1
[αi , βi ]
k∏
j=1
γj = 1, γmj
j = 1〉.
We call (g; m1, . . . , mk ) the signature, and write (m1, . . . , mk ) ifg = 0.
Automorphisms of Arithmetic Riemann Surfaces
![Page 15: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/15.jpg)
Preliminaries The Lower Bound Effective Version
We will use two tools from geometry: the hyperbolic measureon the upper half plane and the Riemann-Hurwitz formula.
Automorphisms of Arithmetic Riemann Surfaces
![Page 16: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/16.jpg)
Preliminaries The Lower Bound Effective Version
We will use two tools from geometry: the hyperbolic measureon the upper half plane and the Riemann-Hurwitz formula.
Theorem (Riemann-Hurwitz)
Recall that the Euler characteristic of a Riemann surface M isdefined in terms of its genus g by χ(M) = 2 − 2g.If f : M → N has degree n, and if ef (P) is the ramificationnumber at P ∈ M, then
χ(N) = nχ(M) +∑
P∈M
(ef (P) − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 17: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/17.jpg)
Preliminaries The Lower Bound Effective Version
We define µ(Γ) to be the hyperbolic measure of Γ\H,
µ(Γ) = µ(g; m1, . . . , mk ) = 2π
2g − 2 +
k∑
j=1
(1 −1mj
)
.
Automorphisms of Arithmetic Riemann Surfaces
![Page 18: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/18.jpg)
Preliminaries The Lower Bound Effective Version
We define µ(Γ) to be the hyperbolic measure of Γ\H,
µ(Γ) = µ(g; m1, . . . , mk ) = 2π
2g − 2 +
k∑
j=1
(1 −1mj
)
.
Using Riemann-Hurwitz, we can show that if Γ′ ≤ Γ is of finiteindex, then
µ(Γ′) = [Γ : Γ′] · µ(Γ).
Automorphisms of Arithmetic Riemann Surfaces
![Page 19: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/19.jpg)
Preliminaries The Lower Bound Effective Version
Consider a Riemann surface as a quotient of H by its surfacegroup.
S = ΓS\H
Automorphisms of Arithmetic Riemann Surfaces
![Page 20: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/20.jpg)
Preliminaries The Lower Bound Effective Version
Consider a Riemann surface as a quotient of H by its surfacegroup.
S = ΓS\H
Then its automorphisms can be obtained from theautomorphisms of H:
Aut(S) = {α ∈ PSL(2, R) : αΓSα−1 = ΓS}/ΓS
= N(ΓS)/ΓS
(Think: Given γ ∈ ΓS , we need α(γ(x)) = γ′(α(x)) for someγ′ ∈ ΓS .)
Automorphisms of Arithmetic Riemann Surfaces
![Page 21: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/21.jpg)
Preliminaries The Lower Bound Effective Version
Given arithmetic Γ, we will build an arithmetic Riemann surfaceS with surface group ΓS , such that Γ ≤ N(ΓS).
Automorphisms of Arithmetic Riemann Surfaces
![Page 22: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/22.jpg)
Preliminaries The Lower Bound Effective Version
Given arithmetic Γ, we will build an arithmetic Riemann surfaceS with surface group ΓS , such that Γ ≤ N(ΓS).
Find a torsion-free normal subgroup K finite index in Γ:
1 // K // Γp // G // 1
Automorphisms of Arithmetic Riemann Surfaces
![Page 23: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/23.jpg)
Preliminaries The Lower Bound Effective Version
Given arithmetic Γ, we will build an arithmetic Riemann surfaceS with surface group ΓS , such that Γ ≤ N(ΓS).
Find a torsion-free normal subgroup K finite index in Γ:
1 // K // Γp // G // 1
Then, if we determine S by ΓS = K , we have
1 // ΓS // N(ΓS) // Aut(S) // 1
1 // ΓS // Γ //?�
OO
G //?�
OO
1
Automorphisms of Arithmetic Riemann Surfaces
![Page 24: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/24.jpg)
Preliminaries The Lower Bound Effective Version
Given arithmetic Γ, we will build an arithmetic Riemann surfaceS with surface group ΓS , such that Γ ≤ N(ΓS).
Find a torsion-free normal subgroup K finite index in Γ:
1 // K // Γp // G // 1
Then, if we determine S by ΓS = K , we have
1 // ΓS // N(ΓS) // Aut(S) // 1
1 // ΓS // Γ //?�
OO
G //?�
OO
1
We call this a surface-kernel epimorphism or SKE.
Automorphisms of Arithmetic Riemann Surfaces
![Page 25: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/25.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
Automorphisms of Arithmetic Riemann Surfaces
![Page 26: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/26.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
For Fuchsian groups, it suffices to check this for the elementsγ1, . . . , γk in the canonical presentation.
Automorphisms of Arithmetic Riemann Surfaces
![Page 27: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/27.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
For Fuchsian groups, it suffices to check this for the elementsγ1, . . . , γk in the canonical presentation.
Given Γ, to build an SKE, need:
Automorphisms of Arithmetic Riemann Surfaces
![Page 28: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/28.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
For Fuchsian groups, it suffices to check this for the elementsγ1, . . . , γk in the canonical presentation.
Given Γ, to build an SKE, need:
• epimorphism p : Γ → G to finite group
Automorphisms of Arithmetic Riemann Surfaces
![Page 29: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/29.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
For Fuchsian groups, it suffices to check this for the elementsγ1, . . . , γk in the canonical presentation.
Given Γ, to build an SKE, need:
• epimorphism p : Γ → G to finite group
• p preserves orders of γi
Automorphisms of Arithmetic Riemann Surfaces
![Page 30: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/30.jpg)
Preliminaries The Lower Bound Effective Version
To verify that the kernel is torsion free, we must check thatevery element of Γ of finite order has its order preserved byp : Γ → G.
For Fuchsian groups, it suffices to check this for the elementsγ1, . . . , γk in the canonical presentation.
Given Γ, to build an SKE, need:
• epimorphism p : Γ → G to finite group
• p preserves orders of γi
Then we know that G is a subgroup of Aut(S).
Automorphisms of Arithmetic Riemann Surfaces
![Page 31: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/31.jpg)
Preliminaries The Lower Bound Effective Version
All triangle groups with a given signature are conjugate, hencetriangle groups with a given signature are either all arithmetic,or none are arithmetic.
Automorphisms of Arithmetic Riemann Surfaces
![Page 32: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/32.jpg)
Preliminaries The Lower Bound Effective Version
All triangle groups with a given signature are conjugate, hencetriangle groups with a given signature are either all arithmetic,or none are arithmetic.
Arithmetic:
(2, 3, n), n = 7, 8, 9, 10, 11, 12, 14, 16, 18, 24, 30
(2, 4, n), n = 5, 6, 7, 8, 9, 10, 12, 18
(2, 5, n), n = 5, 6, 8, 10, 20, 30
etc.
K. Takeuchi. Arithmetic triangle groups. J. Math. Soc. Japan 29(1977), 91-106.
Automorphisms of Arithmetic Riemann Surfaces
![Page 33: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/33.jpg)
Preliminaries The Lower Bound Effective Version
Consider the right-angled hyperbolic pentagon:
π2
π2
π2
π2
π2
Automorphisms of Arithmetic Riemann Surfaces
![Page 34: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/34.jpg)
Preliminaries The Lower Bound Effective Version
Consider the right-angled hyperbolic pentagon:
π2
π2
π2
π2
π2
Let Γ be the orientation-preserving subgroup of the group ofreflections in its sides.
Automorphisms of Arithmetic Riemann Surfaces
![Page 35: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/35.jpg)
Preliminaries The Lower Bound Effective Version
The fundamental domain for Γ is two copies of the pentagon:
Automorphisms of Arithmetic Riemann Surfaces
![Page 36: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/36.jpg)
Preliminaries The Lower Bound Effective Version
The fundamental domain for Γ is two copies of the pentagon:
• Only sequences of an even number of reflections areorientation preserving automorphisms.
Automorphisms of Arithmetic Riemann Surfaces
![Page 37: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/37.jpg)
Preliminaries The Lower Bound Effective Version
The fundamental domain for Γ is two copies of the pentagon:
• Only sequences of an even number of reflections areorientation preserving automorphisms.
• Two reflections give rotation around an angle of π. This isorder 2. There are five such elements of Γ.
Automorphisms of Arithmetic Riemann Surfaces
![Page 38: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/38.jpg)
Preliminaries The Lower Bound Effective Version
The fundamental domain for Γ is two copies of the pentagon:
• Only sequences of an even number of reflections areorientation preserving automorphisms.
• Two reflections give rotation around an angle of π. This isorder 2. There are five such elements of Γ.
• The signature of the group Γ is (2, 2, 2, 2, 2).
Automorphisms of Arithmetic Riemann Surfaces
![Page 39: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/39.jpg)
Preliminaries The Lower Bound Effective Version
The fundamental domain for Γ is two copies of the pentagon:
• Only sequences of an even number of reflections areorientation preserving automorphisms.
• Two reflections give rotation around an angle of π. This isorder 2. There are five such elements of Γ.
• The signature of the group Γ is (2, 2, 2, 2, 2).
• The Riemann surface S = Γ\H is of genus zero.
Automorphisms of Arithmetic Riemann Surfaces
![Page 40: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/40.jpg)
Preliminaries The Lower Bound Effective Version
Subdivide the pentagon into 10 congruent triangles:
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))))))))))
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))
))))
))))
π2
))))
))))
)
���������
HHHHHHHHH
vvvvvvvvv
)))))))))
���������
HHHHHHHHH
vvvvvvvvv
π4
π5
vvvvvvvvvvvv
vvvvvvvvvvvv
To show Γ is arithmetic:
Automorphisms of Arithmetic Riemann Surfaces
![Page 41: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/41.jpg)
Preliminaries The Lower Bound Effective Version
Subdivide the pentagon into 10 congruent triangles:
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))))))))))
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))
))))
))))
π2
))))
))))
)
���������
HHHHHHHHH
vvvvvvvvv
)))))))))
���������
HHHHHHHHH
vvvvvvvvv
π4
π5
vvvvvvvvvvvv
vvvvvvvvvvvv
To show Γ is arithmetic:
• Consider the Fuchsian group Γ′ for a triangle.
Automorphisms of Arithmetic Riemann Surfaces
![Page 42: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/42.jpg)
Preliminaries The Lower Bound Effective Version
Subdivide the pentagon into 10 congruent triangles:
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))))))))))
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))
))))
))))
π2
))))
))))
)
���������
HHHHHHHHH
vvvvvvvvv
)))))))))
���������
HHHHHHHHH
vvvvvvvvv
π4
π5
vvvvvvvvvvvv
vvvvvvvvvvvv
To show Γ is arithmetic:
• Consider the Fuchsian group Γ′ for a triangle.
• The triangle has angles π/2, π/4 and π/5. So Γ′ is the(2, 4, 5) triangle group, which is arithmetic.
Automorphisms of Arithmetic Riemann Surfaces
![Page 43: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/43.jpg)
Preliminaries The Lower Bound Effective Version
Subdivide the pentagon into 10 congruent triangles:
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))))))))))
������������
HHHHHHHHHHHH
vvvvvvvvvvvv
))))
))))
))))
π2
))))
))))
)
���������
HHHHHHHHH
vvvvvvvvv
)))))))))
���������
HHHHHHHHH
vvvvvvvvv
π4
π5
vvvvvvvvvvvv
vvvvvvvvvvvv
To show Γ is arithmetic:
• Consider the Fuchsian group Γ′ for a triangle.
• The triangle has angles π/2, π/4 and π/5. So Γ′ is the(2, 4, 5) triangle group, which is arithmetic.
• But Γ is a subgroup of Γ′ of index 10. Hence the twogroups are commensurable, and so Γ is arithmetic.
Automorphisms of Arithmetic Riemann Surfaces
![Page 44: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/44.jpg)
Preliminaries The Lower Bound Effective Version
Outline
1 Definitions, Geometric Preliminaries and an Example
2 A Sharp Lower Bound on Nar (g)
3 An Effective Version of the Main Theorem
Automorphisms of Arithmetic Riemann Surfaces
![Page 45: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/45.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Let {Sg}g∈G be an infinite sequence of arithmetic surfaces ofdifferent genera g, such that for each g ∈ G, the group ofautomorphisms of Sg has order a(g + b) for some fixed a andb. Then b = −1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 46: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/46.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Let {Sg}g∈G be an infinite sequence of arithmetic surfaces ofdifferent genera g, such that for each g ∈ G, the group ofautomorphisms of Sg has order a(g + b) for some fixed a andb. Then b = −1.
Proof. Let S be a surface from the given sequence.
Then Aut(S) ∼= N(ΓS)/ΓS , where ΓS is the surface groupcorresponding to S.
Automorphisms of Arithmetic Riemann Surfaces
![Page 47: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/47.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Let {Sg}g∈G be an infinite sequence of arithmetic surfaces ofdifferent genera g, such that for each g ∈ G, the group ofautomorphisms of Sg has order a(g + b) for some fixed a andb. Then b = −1.
Proof. Let S be a surface from the given sequence.
Then Aut(S) ∼= N(ΓS)/ΓS , where ΓS is the surface groupcorresponding to S.
The Riemann-Hurwitz formula yields
µ(N(ΓS)) =µ(ΓS)
|Aut(S)|=
2π(2g − 2)
a(g + b),
so µ(N(ΓS)) → 4π/a as g → ∞.
Automorphisms of Arithmetic Riemann Surfaces
![Page 48: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/48.jpg)
Preliminaries The Lower Bound Effective Version
ΓS arithmetic ⇒ N(ΓS) arithmetic.
Automorphisms of Arithmetic Riemann Surfaces
![Page 49: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/49.jpg)
Preliminaries The Lower Bound Effective Version
ΓS arithmetic ⇒ N(ΓS) arithmetic.
The measures of arithmetic groups form a discrete subset of R(Borel).
Automorphisms of Arithmetic Riemann Surfaces
![Page 50: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/50.jpg)
Preliminaries The Lower Bound Effective Version
ΓS arithmetic ⇒ N(ΓS) arithmetic.
The measures of arithmetic groups form a discrete subset of R(Borel).
So for all but finitely many g ∈ G,
2π(2g − 2)
a(g + b)= µ(N(ΓS)) =
4π
a.
Therefore b = −1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 51: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/51.jpg)
Preliminaries The Lower Bound Effective Version
It follows from that the Accola-Maclachlan lower bound forN(g), 8(g + 1), cannot be attained by infinitely many arithmeticsurfaces.
Automorphisms of Arithmetic Riemann Surfaces
![Page 52: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/52.jpg)
Preliminaries The Lower Bound Effective Version
It follows from that the Accola-Maclachlan lower bound forN(g), 8(g + 1), cannot be attained by infinitely many arithmeticsurfaces.
In fact it is never attained by arithmetic surfaces, since theextremal surfaces for this bound are uniformized by surfacesubgroups of (2, 4, 2(g+1))-groups with g ≥ 24 (Maclachlan),and these are not arithmetic (Takeuchi).
Automorphisms of Arithmetic Riemann Surfaces
![Page 53: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/53.jpg)
Preliminaries The Lower Bound Effective Version
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
Automorphisms of Arithmetic Riemann Surfaces
![Page 54: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/54.jpg)
Preliminaries The Lower Bound Effective Version
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
• We prove the inequality by considering a family ofarithmetic surfaces Wg, one for each genus g, and showthat |Aut(Wg)| ≥ 4(g − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 55: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/55.jpg)
Preliminaries The Lower Bound Effective Version
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
• We prove the inequality by considering a family ofarithmetic surfaces Wg, one for each genus g, and showthat |Aut(Wg)| ≥ 4(g − 1).
• We then assume that G := Aut(S) has order |G| > 4(g − 1)for some compact arithmetic surface S of genus g ≥ 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 56: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/56.jpg)
Preliminaries The Lower Bound Effective Version
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
• We prove the inequality by considering a family ofarithmetic surfaces Wg, one for each genus g, and showthat |Aut(Wg)| ≥ 4(g − 1).
• We then assume that G := Aut(S) has order |G| > 4(g − 1)for some compact arithmetic surface S of genus g ≥ 2.
• Imposing specific conditions on g we get a contradiction.
Automorphisms of Arithmetic Riemann Surfaces
![Page 57: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/57.jpg)
Preliminaries The Lower Bound Effective Version
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
• We prove the inequality by considering a family ofarithmetic surfaces Wg, one for each genus g, and showthat |Aut(Wg)| ≥ 4(g − 1).
• We then assume that G := Aut(S) has order |G| > 4(g − 1)for some compact arithmetic surface S of genus g ≥ 2.
• Imposing specific conditions on g we get a contradiction.
• We show that infinitely many values of g satisfy theseconditions. For these g, Nar(g) = 4(g − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 58: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/58.jpg)
Preliminaries The Lower Bound Effective Version
Proof. Let Γ = 〈γ1, . . . , γ5 | γ2j = γ1 . . . γ5 = 1〉 be an arithmetic
group with signature (2, 2, 2, 2, 2).
Automorphisms of Arithmetic Riemann Surfaces
![Page 59: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/59.jpg)
Preliminaries The Lower Bound Effective Version
Proof. Let Γ = 〈γ1, . . . , γ5 | γ2j = γ1 . . . γ5 = 1〉 be an arithmetic
group with signature (2, 2, 2, 2, 2).
Let D2(g−1) = 〈a, b | a2(g−1) = b2 = (ab)2 = 1〉.
Automorphisms of Arithmetic Riemann Surfaces
![Page 60: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/60.jpg)
Preliminaries The Lower Bound Effective Version
Proof. Let Γ = 〈γ1, . . . , γ5 | γ2j = γ1 . . . γ5 = 1〉 be an arithmetic
group with signature (2, 2, 2, 2, 2).
Let D2(g−1) = 〈a, b | a2(g−1) = b2 = (ab)2 = 1〉.
Define θg : Γ → D2(g−1) by γj 7→ ab, b, ag−2b, b, ag−1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 61: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/61.jpg)
Preliminaries The Lower Bound Effective Version
Proof. Let Γ = 〈γ1, . . . , γ5 | γ2j = γ1 . . . γ5 = 1〉 be an arithmetic
group with signature (2, 2, 2, 2, 2).
Let D2(g−1) = 〈a, b | a2(g−1) = b2 = (ab)2 = 1〉.
Define θg : Γ → D2(g−1) by γj 7→ ab, b, ag−2b, b, ag−1.
θg is a SKE and thus Kg = ker(θg) is a surface group.
Automorphisms of Arithmetic Riemann Surfaces
![Page 62: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/62.jpg)
Preliminaries The Lower Bound Effective Version
The surface Wg = Kg\H is arithmetic andAut(Wg) ≥ Kg\Γ ∼= D2(g−1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 63: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/63.jpg)
Preliminaries The Lower Bound Effective Version
The surface Wg = Kg\H is arithmetic andAut(Wg) ≥ Kg\Γ ∼= D2(g−1).
µ(Γ) = π and |D2(g−1)| = 4(g − 1), so by Riemann-Hurwitz
µ(Kg) = µ(Γ)|D2(g−1) = 2π(2g − 2).
Automorphisms of Arithmetic Riemann Surfaces
![Page 64: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/64.jpg)
Preliminaries The Lower Bound Effective Version
The surface Wg = Kg\H is arithmetic andAut(Wg) ≥ Kg\Γ ∼= D2(g−1).
µ(Γ) = π and |D2(g−1)| = 4(g − 1), so by Riemann-Hurwitz
µ(Kg) = µ(Γ)|D2(g−1) = 2π(2g − 2).
Thus Wg has genus g as Wg is a surface group.
Automorphisms of Arithmetic Riemann Surfaces
![Page 65: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/65.jpg)
Preliminaries The Lower Bound Effective Version
The surface Wg = Kg\H is arithmetic andAut(Wg) ≥ Kg\Γ ∼= D2(g−1).
µ(Γ) = π and |D2(g−1)| = 4(g − 1), so by Riemann-Hurwitz
µ(Kg) = µ(Γ)|D2(g−1) = 2π(2g − 2).
Thus Wg has genus g as Wg is a surface group.
Then Nar (g) ≥ |Aut(Wg)| ≥ |D2(g−1)| = 4(g − 1) as required.
Automorphisms of Arithmetic Riemann Surfaces
![Page 66: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/66.jpg)
Preliminaries The Lower Bound Effective Version
Outline of proof that the bound is strict
• Only finitely many signatures with µ(Γ) allowing|K\Γ| = 4(g − 1).
• We set p = g − 1 prime and big enough, based on thesesignatures.
• Then we have a p-Sylow subgroup, which we lift to ∆ ≤ Γand set Q = ∆\Γ.
• T := ∆\H has genus 2 and Q ⊂ Aut(T ).• We have a faithful action of Q on H1(T , Fp).• It decomposes into 1-dimensional submodules.• We find Q ⊂ GL1(Fp)4, which constrains the exponent ǫ of
Q.• Thus ǫ divides gcd(E , p − 1), which we can force to be 2.• This gives a contradiction using the area formula.• We have infinitely many p satisfying our conditions.
Automorphisms of Arithmetic Riemann Surfaces
![Page 67: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/67.jpg)
Preliminaries The Lower Bound Effective Version
We now assume that G ∼= K\Γ for some co-compact arithmeticgroup Γ and normal surface subgroup K = ΓS ≤ Γ, with
4π(g − 1) = µ(K ) = |G|µ(Γ) > 4(g − 1)µ(Γ), (1)
Automorphisms of Arithmetic Riemann Surfaces
![Page 68: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/68.jpg)
Preliminaries The Lower Bound Effective Version
We now assume that G ∼= K\Γ for some co-compact arithmeticgroup Γ and normal surface subgroup K = ΓS ≤ Γ, with
4π(g − 1) = µ(K ) = |G|µ(Γ) > 4(g − 1)µ(Γ), (1)
Borel’s discreteness theorem implies that there are only finitelymany measures of co-compact arithmetic groups µ(Γ) < π.
Automorphisms of Arithmetic Riemann Surfaces
![Page 69: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/69.jpg)
Preliminaries The Lower Bound Effective Version
We now assume that G ∼= K\Γ for some co-compact arithmeticgroup Γ and normal surface subgroup K = ΓS ≤ Γ, with
4π(g − 1) = µ(K ) = |G|µ(Γ) > 4(g − 1)µ(Γ), (1)
Borel’s discreteness theorem implies that there are only finitelymany measures of co-compact arithmetic groups µ(Γ) < π.
Riemann-Hurwitz and (1) show that these correspond to a finiteset Σ of signatures.
Automorphisms of Arithmetic Riemann Surfaces
![Page 70: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/70.jpg)
Preliminaries The Lower Bound Effective Version
We now assume that G ∼= K\Γ for some co-compact arithmeticgroup Γ and normal surface subgroup K = ΓS ≤ Γ, with
4π(g − 1) = µ(K ) = |G|µ(Γ) > 4(g − 1)µ(Γ), (1)
Borel’s discreteness theorem implies that there are only finitelymany measures of co-compact arithmetic groups µ(Γ) < π.
Riemann-Hurwitz and (1) show that these correspond to a finiteset Σ of signatures.
For each σ ∈ Σ, the number
µ(Γ)
4π=
g − 1|G|
=: q =rσsσ
is rational and depends only on the signature σ ∈ Σ. We have|G| = (g − 1)/q = (g − 1)s/r .
Automorphisms of Arithmetic Riemann Surfaces
![Page 71: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/71.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Automorphisms of Arithmetic Riemann Surfaces
![Page 72: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/72.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Let Π denote the finite set of primes which divide an ellipticperiod mj of some signature σ ∈ Σ with rσ = 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 73: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/73.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Let Π denote the finite set of primes which divide an ellipticperiod mj of some signature σ ∈ Σ with rσ = 1.
Let p be a prime such that p /∈ Π, (p, R) = 1 and p > S.Suppose g = p + 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 74: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/74.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Let Π denote the finite set of primes which divide an ellipticperiod mj of some signature σ ∈ Σ with rσ = 1.
Let p be a prime such that p /∈ Π, (p, R) = 1 and p > S.Suppose g = p + 1.
Then |G| = ps with (s, p) = 1 and s < p + 1. By Sylow’sTheorems there is a P ∼= Z/pZ with P E G.
Automorphisms of Arithmetic Riemann Surfaces
![Page 75: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/75.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Let Π denote the finite set of primes which divide an ellipticperiod mj of some signature σ ∈ Σ with rσ = 1.
Let p be a prime such that p /∈ Π, (p, R) = 1 and p > S.Suppose g = p + 1.
Then |G| = ps with (s, p) = 1 and s < p + 1. By Sylow’sTheorems there is a P ∼= Z/pZ with P E G.
Let ∆ denote the inverse image of P in Γ, a normal subgroup ofΓ with Γ/∆ ∼= Q := G/P.
Automorphisms of Arithmetic Riemann Surfaces
![Page 76: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/76.jpg)
Preliminaries The Lower Bound Effective Version
Let R = lcm{rσ|σ ∈ Σ}, and S = max{sσ |σ ∈ Σ, rσ = 1}.
Let Π denote the finite set of primes which divide an ellipticperiod mj of some signature σ ∈ Σ with rσ = 1.
Let p be a prime such that p /∈ Π, (p, R) = 1 and p > S.Suppose g = p + 1.
Then |G| = ps with (s, p) = 1 and s < p + 1. By Sylow’sTheorems there is a P ∼= Z/pZ with P E G.
Let ∆ denote the inverse image of P in Γ, a normal subgroup ofΓ with Γ/∆ ∼= Q := G/P.
Since |Q| is coprime to p, the natural epimorphism G → Qpreserves the orders of the images of all elliptic generators of Γ.
Automorphisms of Arithmetic Riemann Surfaces
![Page 77: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/77.jpg)
Preliminaries The Lower Bound Effective Version
The inclusions K E ∆ E Γ induce an etale Z/pZ-covering ofRiemann surfaces
S ∼= K\H
P∼=Z/pZ
��G
zz
T ∼= ∆\H
Q��
Γ\H
Automorphisms of Arithmetic Riemann Surfaces
![Page 78: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/78.jpg)
Preliminaries The Lower Bound Effective Version
The inclusions K E ∆ E Γ induce an etale Z/pZ-covering ofRiemann surfaces
S ∼= K\H
P∼=Z/pZ
��G
zz
T ∼= ∆\H
Q��
Γ\H
In particular we have that Q ≤ Aut(T ), and T has genus1 + (g − 1)/p = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 79: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/79.jpg)
Preliminaries The Lower Bound Effective Version
The inclusions K E ∆ E Γ induce an etale Z/pZ-covering ofRiemann surfaces
S ∼= K\H
P∼=Z/pZ
��G
zz
T ∼= ∆\H
Q��
Γ\H
In particular we have that Q ≤ Aut(T ), and T has genus1 + (g − 1)/p = 2.
Then Q is contained in a group of automorphisms of aRiemann surface T of genus 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 80: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/80.jpg)
Preliminaries The Lower Bound Effective Version
Notice that |Aut(T )| ≤ 84, thus there are just finitely manypossibilities for Aut(T ) and hence for Q.
Automorphisms of Arithmetic Riemann Surfaces
![Page 81: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/81.jpg)
Preliminaries The Lower Bound Effective Version
Notice that |Aut(T )| ≤ 84, thus there are just finitely manypossibilities for Aut(T ) and hence for Q.
Let E be the least common multiple of the exponents of all thegroups of automorphisms of Riemann surfaces of genus 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 82: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/82.jpg)
Preliminaries The Lower Bound Effective Version
Notice that |Aut(T )| ≤ 84, thus there are just finitely manypossibilities for Aut(T ) and hence for Q.
Let E be the least common multiple of the exponents of all thegroups of automorphisms of Riemann surfaces of genus 2.
Riemann surfaces of genus 2 are hyperelliptic, therefore theirautomorphism groups always contain an element of order 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 83: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/83.jpg)
Preliminaries The Lower Bound Effective Version
Notice that |Aut(T )| ≤ 84, thus there are just finitely manypossibilities for Aut(T ) and hence for Q.
Let E be the least common multiple of the exponents of all thegroups of automorphisms of Riemann surfaces of genus 2.
Riemann surfaces of genus 2 are hyperelliptic, therefore theirautomorphism groups always contain an element of order 2.
In particular E ≡ 0 (mod 2).
Automorphisms of Arithmetic Riemann Surfaces
![Page 84: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/84.jpg)
Preliminaries The Lower Bound Effective Version
We consider first the module structure of H1(T ).
T has genus 2, so H1(T , Z) ∼= Z4.
Automorphisms of Arithmetic Riemann Surfaces
![Page 85: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/85.jpg)
Preliminaries The Lower Bound Effective Version
We consider first the module structure of H1(T ).
T has genus 2, so H1(T , Z) ∼= Z4.
H0(T , Z) ∼= Z, so Tor(H0(T , Z), G) = 0 for all G.
Automorphisms of Arithmetic Riemann Surfaces
![Page 86: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/86.jpg)
Preliminaries The Lower Bound Effective Version
We consider first the module structure of H1(T ).
T has genus 2, so H1(T , Z) ∼= Z4.
H0(T , Z) ∼= Z, so Tor(H0(T , Z), G) = 0 for all G.
By the Universal Coefficient Theorem,
H1(T , Fp) ∼= H1(T , Z) ⊗ Fp∼= F4
p.
Automorphisms of Arithmetic Riemann Surfaces
![Page 87: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/87.jpg)
Preliminaries The Lower Bound Effective Version
We consider first the module structure of H1(T ).
T has genus 2, so H1(T , Z) ∼= Z4.
H0(T , Z) ∼= Z, so Tor(H0(T , Z), G) = 0 for all G.
By the Universal Coefficient Theorem,
H1(T , Fp) ∼= H1(T , Z) ⊗ Fp∼= F4
p.
We also have
H1(T , C) ∼= H1(T , Z) ⊗ C ∼= C4.
Automorphisms of Arithmetic Riemann Surfaces
![Page 88: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/88.jpg)
Preliminaries The Lower Bound Effective Version
Q acts on T , and thus on H1(T , Fp).
Automorphisms of Arithmetic Riemann Surfaces
![Page 89: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/89.jpg)
Preliminaries The Lower Bound Effective Version
Q acts on T , and thus on H1(T , Fp).
The sequence1 → ∆ → Γ → Q → 1
gives an action of Q on ∆.
Automorphisms of Arithmetic Riemann Surfaces
![Page 90: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/90.jpg)
Preliminaries The Lower Bound Effective Version
Q acts on T , and thus on H1(T , Fp).
The sequence1 → ∆ → Γ → Q → 1
gives an action of Q on ∆.
∆ is the group of deck transformations for T , so ∆ ∼= π1(T ).
Automorphisms of Arithmetic Riemann Surfaces
![Page 91: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/91.jpg)
Preliminaries The Lower Bound Effective Version
Q acts on T , and thus on H1(T , Fp).
The sequence1 → ∆ → Γ → Q → 1
gives an action of Q on ∆.
∆ is the group of deck transformations for T , so ∆ ∼= π1(T ).
Thus ∆/∆′ ∼= H1(T , Z) and ∆/∆′∆p ∼= H1(T , Fp).
Automorphisms of Arithmetic Riemann Surfaces
![Page 92: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/92.jpg)
Preliminaries The Lower Bound Effective Version
Q acts on T , and thus on H1(T , Fp).
The sequence1 → ∆ → Γ → Q → 1
gives an action of Q on ∆.
∆ is the group of deck transformations for T , so ∆ ∼= π1(T ).
Thus ∆/∆′ ∼= H1(T , Z) and ∆/∆′∆p ∼= H1(T , Fp).
In fact, these isomorphisms are Q-equivariant.
Automorphisms of Arithmetic Riemann Surfaces
![Page 93: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/93.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
Automorphisms of Arithmetic Riemann Surfaces
![Page 94: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/94.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
These spaces give complex conjugate representations of Q.
Automorphisms of Arithmetic Riemann Surfaces
![Page 95: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/95.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
These spaces give complex conjugate representations of Q.
After Poincare duality, H1(T , C) decomposes into a pair of twodimensional Q-invariant subspaces.
Automorphisms of Arithmetic Riemann Surfaces
![Page 96: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/96.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
These spaces give complex conjugate representations of Q.
After Poincare duality, H1(T , C) decomposes into a pair of twodimensional Q-invariant subspaces.
Those subspaces must both decompose or both be irreducible.
Automorphisms of Arithmetic Riemann Surfaces
![Page 97: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/97.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
These spaces give complex conjugate representations of Q.
After Poincare duality, H1(T , C) decomposes into a pair of twodimensional Q-invariant subspaces.
Those subspaces must both decompose or both be irreducible.
Since p ∤ |Q|, Maschke’s Theorem gives H1(T , Fp) ∼=⊕
Vi .
Automorphisms of Arithmetic Riemann Surfaces
![Page 98: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/98.jpg)
Preliminaries The Lower Bound Effective Version
H1(T , C) ∼= H1,0(T , C) ⊕ H0,1(T , C).
These spaces give complex conjugate representations of Q.
After Poincare duality, H1(T , C) decomposes into a pair of twodimensional Q-invariant subspaces.
Those subspaces must both decompose or both be irreducible.
Since p ∤ |Q|, Maschke’s Theorem gives H1(T , Fp) ∼=⊕
Vi .
So H1(T , Fp) decomposes into a pair of two dimensionalsubspaces, both irreducible or both reducible.
Automorphisms of Arithmetic Riemann Surfaces
![Page 99: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/99.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
Automorphisms of Arithmetic Riemann Surfaces
![Page 100: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/100.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
∆/K ∼= P ∼= Cp, so K contains ∆′∆p.
Automorphisms of Arithmetic Riemann Surfaces
![Page 101: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/101.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
∆/K ∼= P ∼= Cp, so K contains ∆′∆p.
P is a 1-dimensional Fp-vector space with Q-action.
Automorphisms of Arithmetic Riemann Surfaces
![Page 102: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/102.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
∆/K ∼= P ∼= Cp, so K contains ∆′∆p.
P is a 1-dimensional Fp-vector space with Q-action.
So H1(T , Fp) ∼= ∆/∆′∆p ։ ∆/K ∼= P.
Automorphisms of Arithmetic Riemann Surfaces
![Page 103: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/103.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
∆/K ∼= P ∼= Cp, so K contains ∆′∆p.
P is a 1-dimensional Fp-vector space with Q-action.
So H1(T , Fp) ∼= ∆/∆′∆p ։ ∆/K ∼= P.
Thus
V = H1(T , Fp) ∼=
4⊕
i=1
Vi ,
with each Vi a 1-dimensional Q-invariant subspace of V .
Automorphisms of Arithmetic Riemann Surfaces
![Page 104: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/104.jpg)
Preliminaries The Lower Bound Effective Version
We now construct a 1-dimensional quotient of H1(T , Fp).
∆/K ∼= P ∼= Cp, so K contains ∆′∆p.
P is a 1-dimensional Fp-vector space with Q-action.
So H1(T , Fp) ∼= ∆/∆′∆p ։ ∆/K ∼= P.
Thus
V = H1(T , Fp) ∼=
4⊕
i=1
Vi ,
with each Vi a 1-dimensional Q-invariant subspace of V .
Therefore we have a map Q → GL1(Fp)4.
Automorphisms of Arithmetic Riemann Surfaces
![Page 105: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/105.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Lemma (Farkas & Kra, V.3.4, due to Serre) If A ∈ SLk (Z) hasfinite order m > 1 and A ≡ I (mod n) then m = n = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 106: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/106.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Lemma (Farkas & Kra, V.3.4, due to Serre) If A ∈ SLk (Z) hasfinite order m > 1 and A ≡ I (mod n) then m = n = 2.
So in fact, Q → GL1(Fp)4 ∼= (Cp−1)4.
Automorphisms of Arithmetic Riemann Surfaces
![Page 107: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/107.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Lemma (Farkas & Kra, V.3.4, due to Serre) If A ∈ SLk (Z) hasfinite order m > 1 and A ≡ I (mod n) then m = n = 2.
So in fact, Q → GL1(Fp)4 ∼= (Cp−1)4.
Therefore Q has exponent ǫ dividing p − 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 108: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/108.jpg)
Preliminaries The Lower Bound Effective Version
Lemma
Lemma (Farkas & Kra, V.3.4, due to Serre) If A ∈ SLk (Z) hasfinite order m > 1 and A ≡ I (mod n) then m = n = 2.
So in fact, Q → GL1(Fp)4 ∼= (Cp−1)4.
Therefore Q has exponent ǫ dividing p − 1.
ǫ thus divides gcd(E , p − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 109: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/109.jpg)
Preliminaries The Lower Bound Effective Version
Choose p with gcd(E , p − 1) = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 110: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/110.jpg)
Preliminaries The Lower Bound Effective Version
Choose p with gcd(E , p − 1) = 2.
∆ is a surface group, so each elliptic period equals 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 111: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/111.jpg)
Preliminaries The Lower Bound Effective Version
Choose p with gcd(E , p − 1) = 2.
∆ is a surface group, so each elliptic period equals 2.
Recall our area formula:
µ(Γ) = 2π(2g − 2 +
k∑
i=1
(1 −1mi
)).
If all mi are 2, we must have µ(Γ) a multiple of π.
Automorphisms of Arithmetic Riemann Surfaces
![Page 112: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/112.jpg)
Preliminaries The Lower Bound Effective Version
Choose p with gcd(E , p − 1) = 2.
∆ is a surface group, so each elliptic period equals 2.
Recall our area formula:
µ(Γ) = 2π(2g − 2 +
k∑
i=1
(1 −1mi
)).
If all mi are 2, we must have µ(Γ) a multiple of π.
This contradicts 0 < µ(Γ) < π.
Automorphisms of Arithmetic Riemann Surfaces
![Page 113: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/113.jpg)
Preliminaries The Lower Bound Effective Version
In summary, we have required that g − 1 = p is prime, p > S,p /∈ Π, p is coprime to R and gcd(p − 1, E) = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 114: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/114.jpg)
Preliminaries The Lower Bound Effective Version
In summary, we have required that g − 1 = p is prime, p > S,p /∈ Π, p is coprime to R and gcd(p − 1, E) = 2.
By Dirichlet’s theorem, there are infinitely primes
p ≡ −1 (mod E).
Automorphisms of Arithmetic Riemann Surfaces
![Page 115: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/115.jpg)
Preliminaries The Lower Bound Effective Version
In summary, we have required that g − 1 = p is prime, p > S,p /∈ Π, p is coprime to R and gcd(p − 1, E) = 2.
By Dirichlet’s theorem, there are infinitely primes
p ≡ −1 (mod E).
All but finitely many satisfy the other required properties.
Automorphisms of Arithmetic Riemann Surfaces
![Page 116: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/116.jpg)
Preliminaries The Lower Bound Effective Version
In summary, we have required that g − 1 = p is prime, p > S,p /∈ Π, p is coprime to R and gcd(p − 1, E) = 2.
By Dirichlet’s theorem, there are infinitely primes
p ≡ −1 (mod E).
All but finitely many satisfy the other required properties.
Therefore we have an infinitely many g that lead to acontradiction.
Automorphisms of Arithmetic Riemann Surfaces
![Page 117: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/117.jpg)
Preliminaries The Lower Bound Effective Version
In summary, we have required that g − 1 = p is prime, p > S,p /∈ Π, p is coprime to R and gcd(p − 1, E) = 2.
By Dirichlet’s theorem, there are infinitely primes
p ≡ −1 (mod E).
All but finitely many satisfy the other required properties.
Therefore we have an infinitely many g that lead to acontradiction.
So we’ve proven:
Theorem
Nar(g) ≥ 4(g − 1) for all g ≥ 2, and this bound is attained forinfinitely many values of g.
Automorphisms of Arithmetic Riemann Surfaces
![Page 118: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/118.jpg)
Preliminaries The Lower Bound Effective Version
Outline
1 Definitions, Geometric Preliminaries and an Example
2 A Sharp Lower Bound on Nar (g)
3 An Effective Version of the Main Theorem
Automorphisms of Arithmetic Riemann Surfaces
![Page 119: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/119.jpg)
Preliminaries The Lower Bound Effective Version
Main Theorem
• Main Theorem: Let Σ be the set of all signatures ofcocompact arithmetic Fuchsian groups with volume strictlyless than π.
Automorphisms of Arithmetic Riemann Surfaces
![Page 120: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/120.jpg)
Preliminaries The Lower Bound Effective Version
Main Theorem
• Main Theorem: Let Σ be the set of all signatures ofcocompact arithmetic Fuchsian groups with volume strictlyless than π.Writing µ(Γσ)
4π as a fraction rσ/sσ in lowest terms for everyσ ∈ Σ, let R = lcm{rσ}, let Π be the list of primes thatdivide the period of an elliptic element of one of the Γσ, andS = max{sσ}.
Automorphisms of Arithmetic Riemann Surfaces
![Page 121: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/121.jpg)
Preliminaries The Lower Bound Effective Version
Main Theorem
• Main Theorem: Let Σ be the set of all signatures ofcocompact arithmetic Fuchsian groups with volume strictlyless than π.Writing µ(Γσ)
4π as a fraction rσ/sσ in lowest terms for everyσ ∈ Σ, let R = lcm{rσ}, let Π be the list of primes thatdivide the period of an elliptic element of one of the Γσ, andS = max{sσ}.Assume that g − 1 =: p is a prime such that gcd(p, R) = 1,p 6∈ Π, p > S and such that gcd(p − 1, E) = 2, where E isthe least common multiple of the exponents of allautomorphism groups of Riemann surfaces of genus 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 122: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/122.jpg)
Preliminaries The Lower Bound Effective Version
Main Theorem
• Main Theorem: Let Σ be the set of all signatures ofcocompact arithmetic Fuchsian groups with volume strictlyless than π.Writing µ(Γσ)
4π as a fraction rσ/sσ in lowest terms for everyσ ∈ Σ, let R = lcm{rσ}, let Π be the list of primes thatdivide the period of an elliptic element of one of the Γσ, andS = max{sσ}.Assume that g − 1 =: p is a prime such that gcd(p, R) = 1,p 6∈ Π, p > S and such that gcd(p − 1, E) = 2, where E isthe least common multiple of the exponents of allautomorphism groups of Riemann surfaces of genus 2.Then the size of the automorphism group of any surface ofgenus g cannot be greater than 4(g − 1), so we have tohave equality.
Automorphisms of Arithmetic Riemann Surfaces
![Page 123: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/123.jpg)
Preliminaries The Lower Bound Effective Version
Explicit Sequence Theorem
Goal
Construct a specific sequence of genera g such that Nar attainsthe lower bound.
Automorphisms of Arithmetic Riemann Surfaces
![Page 124: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/124.jpg)
Preliminaries The Lower Bound Effective Version
Explicit Sequence Theorem
Goal
Construct a specific sequence of genera g such that Nar attainsthe lower bound.
Theorem (Explicit Theorem)
For all primes p ≡ 23, 47, 59 (mod 60), we haveNar (g) = 4(g − 1). The least genus g for which the lower boundNar (g) = 4(g − 1) is attained is g = 24.
Automorphisms of Arithmetic Riemann Surfaces
![Page 125: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/125.jpg)
Preliminaries The Lower Bound Effective Version
Explicit Sequence Theorem
Goal
Construct a specific sequence of genera g such that Nar attainsthe lower bound.
Theorem (Explicit Theorem)
For all primes p ≡ 23, 47, 59 (mod 60), we haveNar (g) = 4(g − 1). The least genus g for which the lower boundNar (g) = 4(g − 1) is attained is g = 24.
Idea
Construct primes p satisfying the hypotheses of the MainTheorem. Then g = p + 1 will be such that:
Nar (g) = 4(g − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 126: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/126.jpg)
Preliminaries The Lower Bound Effective Version
Strategy
1 Listing all Arithmetic Fuchsian Signatures2 The Conditions on Sufficiently Large Primes p3 Smaller Primes
Automorphisms of Arithmetic Riemann Surfaces
![Page 127: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/127.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Want to find the set Σ of all signatures of cocompactarithmetic Fuchsian groups with volume strictly less than π.
Automorphisms of Arithmetic Riemann Surfaces
![Page 128: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/128.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Want to find the set Σ of all signatures of cocompactarithmetic Fuchsian groups with volume strictly less than π.
• Writing µ(Γσ) as a fraction rσ/sσ in lowest terms for everyσ ∈ Σ, we need to determine R = lcm{rσ}, the list Π ofprimes that divide an elliptic period mk , and S = max{sσ}.
Automorphisms of Arithmetic Riemann Surfaces
![Page 129: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/129.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Want to find the set Σ of all signatures of cocompactarithmetic Fuchsian groups with volume strictly less than π.
• Writing µ(Γσ) as a fraction rσ/sσ in lowest terms for everyσ ∈ Σ, we need to determine R = lcm{rσ}, the list Π ofprimes that divide an elliptic period mk , and S = max{sσ}.
• Then by the proof of the Main Theorem, for any prime pnot dividing R, not contained in Π and greater than S, wecannot have
|G| > 4(g − 1)
if we impose the additional condition thatgcd(p − 1, E) = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 130: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/130.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Let (g; m1; . . . ; mr ) be the signature of a Fuchsian group Γ.Then
1π
µ(Γ) = 4(g − 1) + 2r
∑
k=1
(
1 −1
mk
)
< 1 (2)
has no solution unless g = 0.
Automorphisms of Arithmetic Riemann Surfaces
![Page 131: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/131.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Let (g; m1; . . . ; mr ) be the signature of a Fuchsian group Γ.Then
1π
µ(Γ) = 4(g − 1) + 2r
∑
k=1
(
1 −1
mk
)
< 1 (2)
has no solution unless g = 0.
• If g = 0, then since mk ≥ 2, we must have r < 5, so allsignatures have length 3 or 4.
Automorphisms of Arithmetic Riemann Surfaces
![Page 132: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/132.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• Let (g; m1; . . . ; mr ) be the signature of a Fuchsian group Γ.Then
1π
µ(Γ) = 4(g − 1) + 2r
∑
k=1
(
1 −1
mk
)
< 1 (2)
has no solution unless g = 0.
• If g = 0, then since mk ≥ 2, we must have r < 5, so allsignatures have length 3 or 4.
• Takeuchi gave a complete list of cocompact arithmetictriangle groups; almost all of these have volume less thanπ.
Automorphisms of Arithmetic Riemann Surfaces
![Page 133: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/133.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• The only other possible candidates are(2, 2, 3, 3),(2, 2, 3, 4),(2, 2, 3, 5) and (2, 2, 2, n), for n ≥ 3.
Automorphisms of Arithmetic Riemann Surfaces
![Page 134: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/134.jpg)
Preliminaries The Lower Bound Effective Version
List of Possible Signatures
• The only other possible candidates are(2, 2, 3, 3),(2, 2, 3, 4),(2, 2, 3, 5) and (2, 2, 2, n), for n ≥ 3.
• It can be shown that there are only 12 signatures for which(2, 2, 2, n) is arithmetic.
Automorphisms of Arithmetic Riemann Surfaces
![Page 135: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/135.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Note that the orders of the elliptic elements are either2,3,4,5 or 7, so Π = {2, 3, 5, 7}.
Automorphisms of Arithmetic Riemann Surfaces
![Page 136: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/136.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Note that the orders of the elliptic elements are either2,3,4,5 or 7, so Π = {2, 3, 5, 7}.
• Further examining the list of possible signatures, andputting µ(Γ)
4π into lowest terms, we find that R = 4 · 3 · 5 · 7 is
the least common multiple of the numerators of all µ(Γσ)4π
and s = 84 is the largest occurring denominator.
Automorphisms of Arithmetic Riemann Surfaces
![Page 137: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/137.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Note that the orders of the elliptic elements are either2,3,4,5 or 7, so Π = {2, 3, 5, 7}.
• Further examining the list of possible signatures, andputting µ(Γ)
4π into lowest terms, we find that R = 4 · 3 · 5 · 7 is
the least common multiple of the numerators of all µ(Γσ)4π
and s = 84 is the largest occurring denominator.
• To deal with the last condition gcd(p − 1, E) = 2, we needa lemma:
Automorphisms of Arithmetic Riemann Surfaces
![Page 138: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/138.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Note that the orders of the elliptic elements are either2,3,4,5 or 7, so Π = {2, 3, 5, 7}.
• Further examining the list of possible signatures, andputting µ(Γ)
4π into lowest terms, we find that R = 4 · 3 · 5 · 7 is
the least common multiple of the numerators of all µ(Γσ)4π
and s = 84 is the largest occurring denominator.
• To deal with the last condition gcd(p − 1, E) = 2, we needa lemma:
Lemma
If S is a Riemann surface of genus γ ≥ 2, then it has noautomorphisms of prime order greater than 2γ + 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 139: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/139.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
Automorphisms of Arithmetic Riemann Surfaces
![Page 140: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/140.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.
Automorphisms of Arithmetic Riemann Surfaces
![Page 141: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/141.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.Then f : S −→ T is a smooth p-sheeted covering of T , so theRiemann-Hurwitz formula reads:
2(γ − 1) = 2p(γ′ − 1) + m(p − 1)
Automorphisms of Arithmetic Riemann Surfaces
![Page 142: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/142.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.Then f : S −→ T is a smooth p-sheeted covering of T , so theRiemann-Hurwitz formula reads:
2(γ − 1) = 2p(γ′ − 1) + m(p − 1)
where m is the number of fixed points of f .
Automorphisms of Arithmetic Riemann Surfaces
![Page 143: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/143.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.Then f : S −→ T is a smooth p-sheeted covering of T , so theRiemann-Hurwitz formula reads:
2(γ − 1) = 2p(γ′ − 1) + m(p − 1)
where m is the number of fixed points of f . Assume that p ≥ 2γ,then
Automorphisms of Arithmetic Riemann Surfaces
![Page 144: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/144.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.Then f : S −→ T is a smooth p-sheeted covering of T , so theRiemann-Hurwitz formula reads:
2(γ − 1) = 2p(γ′ − 1) + m(p − 1)
where m is the number of fixed points of f . Assume that p ≥ 2γ,then
• for γ′ ≥ 2, 2(γ − 1) ≥ 2p + m(p − 1) ≥ 2p, a contradiction
Automorphisms of Arithmetic Riemann Surfaces
![Page 145: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/145.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
Proof.
If f is an automorphism of S of order p, let T be the Riemannsurface corresponding to S modulo < f >, and γ′ its genus.Then f : S −→ T is a smooth p-sheeted covering of T , so theRiemann-Hurwitz formula reads:
2(γ − 1) = 2p(γ′ − 1) + m(p − 1)
where m is the number of fixed points of f . Assume that p ≥ 2γ,then
• for γ′ ≥ 2, 2(γ − 1) ≥ 2p + m(p − 1) ≥ 2p, a contradiction
• for γ′ = 1, 2(γ − 1) = m(p − 1) ≥ p − 1 ≥ 2γ − 1, acontradiction.
Automorphisms of Arithmetic Riemann Surfaces
![Page 146: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/146.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• For γ′ = 0, 2(γ − 1) = −2pg + m(p − 1), we havem = 2γ
p−1 + 2 ≤ pp−1 + 2 ≤ 3, so m = 3.
Automorphisms of Arithmetic Riemann Surfaces
![Page 147: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/147.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• For γ′ = 0, 2(γ − 1) = −2pg + m(p − 1), we havem = 2γ
p−1 + 2 ≤ pp−1 + 2 ≤ 3, so m = 3.
• In this case, 2γ − 2 = −2p + 3(p − 1), so p = 2γ + 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 148: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/148.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• For γ′ = 0, 2(γ − 1) = −2pg + m(p − 1), we havem = 2γ
p−1 + 2 ≤ pp−1 + 2 ≤ 3, so m = 3.
• In this case, 2γ − 2 = −2p + 3(p − 1), so p = 2γ + 1.Hence it follows that p ≤ 2γ + 1.
Automorphisms of Arithmetic Riemann Surfaces
![Page 149: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/149.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• For γ′ = 0, 2(γ − 1) = −2pg + m(p − 1), we havem = 2γ
p−1 + 2 ≤ pp−1 + 2 ≤ 3, so m = 3.
• In this case, 2γ − 2 = −2p + 3(p − 1), so p = 2γ + 1.Hence it follows that p ≤ 2γ + 1.
• So if S is a surface of genus 2, it cannot haveautomorphisms of prime order q for any q > 5. Thus theexponent of Aut(S) is not divisible by any prime other than2,3 or 5.
Automorphisms of Arithmetic Riemann Surfaces
![Page 150: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/150.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• For γ′ = 0, 2(γ − 1) = −2pg + m(p − 1), we havem = 2γ
p−1 + 2 ≤ pp−1 + 2 ≤ 3, so m = 3.
• In this case, 2γ − 2 = −2p + 3(p − 1), so p = 2γ + 1.Hence it follows that p ≤ 2γ + 1.
• So if S is a surface of genus 2, it cannot haveautomorphisms of prime order q for any q > 5. Thus theexponent of Aut(S) is not divisible by any prime other than2,3 or 5.
Automorphisms of Arithmetic Riemann Surfaces
![Page 151: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/151.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Conclusion: No prime other than {2, 3, 5} divides E , theleast common multiple of the exponents of automorphismgroups of surfaces of genus 2. Thus the condition thatgcd(p − 1, E) = 2 is satisfied by all p such that p − 1 is notdivisible by 3, 4, 5.
Automorphisms of Arithmetic Riemann Surfaces
![Page 152: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/152.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently Large Primes
• Conclusion: No prime other than {2, 3, 5} divides E , theleast common multiple of the exponents of automorphismgroups of surfaces of genus 2. Thus the condition thatgcd(p − 1, E) = 2 is satisfied by all p such that p − 1 is notdivisible by 3, 4, 5.
• Since we also require that p 6≡ 0 mod q for q = 2,3,5, thisleaves the possibilities that p ≡ 2 (mod 3), p ≡ 3 mod 4and p ≡ 2, 3, 4 mod 5. The first two lift to the congruencep ≡ 11 (mod 12); combining with the last one givesp ≡ 23, 47, 59 (mod 60) as the equivalent congruence.
Automorphisms of Arithmetic Riemann Surfaces
![Page 153: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/153.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently large Primes/Smaller Primes
• We have shown that any prime p > 84 congruent to one of23,47,59 modulo 60 satisfies the conditions of the MainTheorem.
Automorphisms of Arithmetic Riemann Surfaces
![Page 154: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/154.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently large Primes/Smaller Primes
• We have shown that any prime p > 84 congruent to one of23,47,59 modulo 60 satisfies the conditions of the MainTheorem.
• Thus, surfaces of genus p + 1 for any such p satisfy thelower bound: Ng = 4(g − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 155: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/155.jpg)
Preliminaries The Lower Bound Effective Version
Sufficiently large Primes/Smaller Primes
• We have shown that any prime p > 84 congruent to one of23,47,59 modulo 60 satisfies the conditions of the MainTheorem.
• Thus, surfaces of genus p + 1 for any such p satisfy thelower bound: Ng = 4(g − 1).
• What about p = 23, 47, 59 or 83?
Automorphisms of Arithmetic Riemann Surfaces
![Page 156: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/156.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
Automorphisms of Arithmetic Riemann Surfaces
![Page 157: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/157.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
• 59 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 59s forsome s.
Automorphisms of Arithmetic Riemann Surfaces
![Page 158: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/158.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
• 59 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 59s forsome s.
• By inspection, s is coprime to 59, so a 59-Sylow subgroupis of order 59. Letting n59 be the number of 59-Sylowsubgroups, we must have n59|s and n59 ≡ 1(mod 59) ⇒ n59 = 1. So the 59-Sylow subgroup P59 isunique.
Automorphisms of Arithmetic Riemann Surfaces
![Page 159: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/159.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
• 59 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 59s forsome s.
• By inspection, s is coprime to 59, so a 59-Sylow subgroupis of order 59. Letting n59 be the number of 59-Sylowsubgroups, we must have n59|s and n59 ≡ 1(mod 59) ⇒ n59 = 1. So the 59-Sylow subgroup P59 isunique.
• p 6∈ Π = {2, 3, 5, 7}, the set of primes dividing an elementof order in some Γσ.
Automorphisms of Arithmetic Riemann Surfaces
![Page 160: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/160.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
• 59 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 59s forsome s.
• By inspection, s is coprime to 59, so a 59-Sylow subgroupis of order 59. Letting n59 be the number of 59-Sylowsubgroups, we must have n59|s and n59 ≡ 1(mod 59) ⇒ n59 = 1. So the 59-Sylow subgroup P59 isunique.
• p 6∈ Π = {2, 3, 5, 7}, the set of primes dividing an elementof order in some Γσ.
• p − 1 = 58 = 2 · 29, so gcd(p − 1, E) = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 161: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/161.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=59
• p = 59, S of genus g = 60:
• 59 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 59s forsome s.
• By inspection, s is coprime to 59, so a 59-Sylow subgroupis of order 59. Letting n59 be the number of 59-Sylowsubgroups, we must have n59|s and n59 ≡ 1(mod 59) ⇒ n59 = 1. So the 59-Sylow subgroup P59 isunique.
• p 6∈ Π = {2, 3, 5, 7}, the set of primes dividing an elementof order in some Γσ.
• p − 1 = 58 = 2 · 29, so gcd(p − 1, E) = 2.
• Conclusion: g = 60 attains the lower bound.
Automorphisms of Arithmetic Riemann Surfaces
![Page 162: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/162.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:
Automorphisms of Arithmetic Riemann Surfaces
![Page 163: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/163.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:• 83 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 83s for
some s. By inspection, s is coprime to 83, so if P83 is a83-Sylow subgroup, then |P83| = 83. Letting n83 be thenumber of 83-Sylow subgroups, we must have n83|s andn59 ≡ 1 (mod 59).
Automorphisms of Arithmetic Riemann Surfaces
![Page 164: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/164.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:• 83 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 83s for
some s. By inspection, s is coprime to 83, so if P83 is a83-Sylow subgroup, then |P83| = 83. Letting n83 be thenumber of 83-Sylow subgroups, we must have n83|s andn59 ≡ 1 (mod 59).
• Claim: P83 is normal in G.
Automorphisms of Arithmetic Riemann Surfaces
![Page 165: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/165.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:• 83 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 83s for
some s. By inspection, s is coprime to 83, so if P83 is a83-Sylow subgroup, then |P83| = 83. Letting n83 be thenumber of 83-Sylow subgroups, we must have n83|s andn59 ≡ 1 (mod 59).
• Claim: P83 is normal in G.
Proof:
Automorphisms of Arithmetic Riemann Surfaces
![Page 166: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/166.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:• 83 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 83s for
some s. By inspection, s is coprime to 83, so if P83 is a83-Sylow subgroup, then |P83| = 83. Letting n83 be thenumber of 83-Sylow subgroups, we must have n83|s andn59 ≡ 1 (mod 59).
• Claim: P83 is normal in G.
Proof:• The only possibility for the 83-Sylow subgroup P83 not
being unique is if n83 = s = 84.
Automorphisms of Arithmetic Riemann Surfaces
![Page 167: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/167.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83
• p = 83, S of genus g = 84:• 83 is coprime to R, so |Aut(S)| = |G| = (g − 1)s = 83s for
some s. By inspection, s is coprime to 83, so if P83 is a83-Sylow subgroup, then |P83| = 83. Letting n83 be thenumber of 83-Sylow subgroups, we must have n83|s andn59 ≡ 1 (mod 59).
• Claim: P83 is normal in G.
Proof:• The only possibility for the 83-Sylow subgroup P83 not
being unique is if n83 = s = 84.• Then the normaliser of P83 is just P, so G acts faithfully
and transitively on P83 (Frobenius action).⇒ There exists a normal subgroup N of G such that G isthe semidirect product of N and P83.
Automorphisms of Arithmetic Riemann Surfaces
![Page 168: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/168.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.
Automorphisms of Arithmetic Riemann Surfaces
![Page 169: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/169.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.
Automorphisms of Arithmetic Riemann Surfaces
![Page 170: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/170.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2.
Automorphisms of Arithmetic Riemann Surfaces
![Page 171: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/171.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2. Therefore,
p = 83 satisfies all required conditions to exclude that|G| > 4(g − 1) = 4 · 83.
Automorphisms of Arithmetic Riemann Surfaces
![Page 172: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/172.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2. Therefore,
p = 83 satisfies all required conditions to exclude that|G| > 4(g − 1) = 4 · 83.
• Conclusion: g = 84 attains the lower bound.
Automorphisms of Arithmetic Riemann Surfaces
![Page 173: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/173.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2. Therefore,
p = 83 satisfies all required conditions to exclude that|G| > 4(g − 1) = 4 · 83.
• Conclusion: g = 84 attains the lower bound.• Similarly, one can show that for p = g − 1 = 47, there
exists a unique normal subgroup of order 47, and satisfiesthe other conditions of the Main Theorem as well.
Automorphisms of Arithmetic Riemann Surfaces
![Page 174: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/174.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2. Therefore,
p = 83 satisfies all required conditions to exclude that|G| > 4(g − 1) = 4 · 83.
• Conclusion: g = 84 attains the lower bound.• Similarly, one can show that for p = g − 1 = 47, there
exists a unique normal subgroup of order 47, and satisfiesthe other conditions of the Main Theorem as well.
• Using more results from group theory, one can show thatp = 23 attains the lower bound as well.
Automorphisms of Arithmetic Riemann Surfaces
![Page 175: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/175.jpg)
Preliminaries The Lower Bound Effective Version
Smaller Primes: p=83,47,23
• In particular, there exists an epimorphism G → Z83.• But since s = 84, Γ = Γ(2, 3, 7) is a triangle group, this is
impossible. Thus P83 must be normal as required.• Also, p − 1 = 82 = 2 · 41, so gcd(p − 1, E) = 2. Therefore,
p = 83 satisfies all required conditions to exclude that|G| > 4(g − 1) = 4 · 83.
• Conclusion: g = 84 attains the lower bound.• Similarly, one can show that for p = g − 1 = 47, there
exists a unique normal subgroup of order 47, and satisfiesthe other conditions of the Main Theorem as well.
• Using more results from group theory, one can show thatp = 23 attains the lower bound as well.
• In fact, one can show that g = 24 is the smallest genussuch that Nar (g) = 4(g − 1).
Automorphisms of Arithmetic Riemann Surfaces
![Page 176: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/176.jpg)
Preliminaries The Lower Bound Effective Version
Explicit Sequence Theorem
Theorem (Explicit Theorem)
For all primes p ≡ 23, 47, 59 (mod 6)0, we haveNar (g) = 4(g − 1). The least genus g for which the the lowerbound Nar (g) = 4(g − 1) is attained is g = 24.
Automorphisms of Arithmetic Riemann Surfaces
![Page 177: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/177.jpg)
Preliminaries The Lower Bound Effective Version
References I
Farkas, H.M., Kra, I.Riemann Surfaces.New York: Springer, 1980.
Jones, G., Singerman, D.Complex Functions.Cambridge: Cambridge UP, 1987.
Katok, S.Fuchsian Groups.Chicago: U Chicago Press, 1992.
Automorphisms of Arithmetic Riemann Surfaces
![Page 178: A bound for the number of automorphisms of an arithmetic …math.mit.edu/~roed/writings/talks/mAWS08.pdf · 2017-11-08 · Preliminaries The Lower Bound Effective Version Definition](https://reader033.fdocuments.in/reader033/viewer/2022042802/5f3fd6e9b8ef361ab9456304/html5/thumbnails/178.jpg)
Preliminaries The Lower Bound Effective Version
References II
Belolipetsky, M., Jones, G.A bound for the number of automorphisms of an arithmeticRiemann surface.Math. Proc. Cambridge Philos. Soc., 138 (2005), no. 2,289-299.
Sah, C.H.Groups related to compact Riemann surfaces.Acta Math. 123 (1969), 13-42.
Takeuchi, K.Arithmetic Triangle GroupsJ. Math. Soc. Japan 29 (1977), 91-106.
Automorphisms of Arithmetic Riemann Surfaces