From Klein's Platonic Solids to Kepler's Archimedean Solids: [3pt ...

Dessin d’EnfantsExamples due to Magot and Zvonkin

Moduli Spaces

From Klein’s Platonic Solidsto Kepler’s Archimedean Solids:

Elliptic Curves and Dessins d’Enfants

Part II

Edray Herber Goins

Department of MathematicsPurdue University

September 7, 2012

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids


Moduli Spaces

Abstract

In 1884, Felix Klein wrote his influential book, “Lectures on theIcosahedron,” where he explained how to express the roots of anyquintic polynomial in terms of elliptic modular functions. His idea wasto relate rotations of the icosahedron with the automorphism groupof 5-torsion points on a suitable elliptic curve. In fact, he created atheory which related rotations of each of the five regular solids (thetetrahedron, cube, octahedron, icosahedron, and dodecahedron) withthe automorphism groups of 3-, 4-, and 5-torsion points.

Using modern language, the functions which relate the rotations withelliptic curves are Belyı maps. In 1984, Alexander Grothendieckintroduced the concept of a Dessin d’Enfant in order to understandGalois groups via such maps. We will complete a circle of ideas byreviewing Klein’s theory with an emphasis on the octahedron;explaining how to realize the five regular solids (the Platonic solids)as well as the thirteen semi-regular solids (the Archimedean solids) asDessins d’Enfant; and discussing how the corresponding Belyı mapsrelate to moduli spaces of elliptic curves.



Moduli Spaces

Outline of Talk

1 Dessin d’EnfantsPlatonic SolidsKlein’s Theory of CovariantsBelyı’s Theorem

2 Examples due to Magot and ZvonkinRotation Group Dn

Rotation Groups A4 and S4

Rotation Group A5

3 Moduli SpacesX (3), X (4), and X (5)X0(2), X (2), and X (2, 4)Torsion on Elliptic Curves



Moduli Spaces

Platonic SolidsKlein’s Theory of CovariantsBelyı’s Theorem

Recap



Moduli Spaces


What is a Platonic Solid?

A regular, convex polyhedron is one of the collections of vertices

V =

{(z1 : z0) ∈ P1(C)

∣∣∣∣ δ(z1, z0) = 0

}in terms of the homogeneous polynomials

δ(z1, z0) =

z1n + z0

n for the regular polygon,

z1(z1

3 − z03)

for the tetrahedron,

z1 z0(z1

4 − z04)

for the octahedron,

z18 + 14 z1

4 z04 + z0

8 for the cube,

z1 z0(z1

10 − 11 z15 z0

5 − z010)

for the icosahedron,

z120 + 228 z1

15 z05 + 494 z1

10 z010

− 228 z15 z0

15 + z020

for the dodecahedron.

We embed V ↪→ S2(R) into the unit sphere via stereographic projection.



Moduli Spaces


http://mathworld.wolfram.com/RegularPolygon.html

http://en.wikipedia.org/wiki/Platonic_solids


http://mathworld.wolfram.com/RegularPolygon.html

http://en.wikipedia.org/wiki/Platonic_solids


Moduli Spaces


Rigid Rotations of the Platonic Solids

Recall the action ◦ : PSL2(C)× P1(C)→ P1(C). We seek Galois extensionsQ(ζn, z)/Q(ζn, j) for rational j(z).

Zn =⟨r∣∣ rn = 1

⟩and Dn =

⟨r , s

∣∣ s2 = rn = (s r)2 = 1⟩

are the rigidrotations of the regular convex polygons, with

r(z) = ζn z, s(z) =1

z, and j(z) = 1728 zn or

6912 zn

(zn + 1)2.

A4 =⟨r , s

∣∣ s2 = r 3 = (s r)3 = 1⟩' PSL2(F3) are the rigid rotations of

the tetrahedron, with

r(z) = ζ3 z, s(z) =1 − z

2 z + 1, and j(z) = −

27 (8 z3 + 1)3

z3 (z3 − 1)3.

S4 =⟨r , s

∣∣ s2 = r 3 = (s r)4 = 1⟩' PGL2(F3) ' PSL2(Z/4Z) are the

rigid rotations of the octahedron and the cube, with

r(z) =ζ4 + z

ζ4 − z, s(z) =

1 − z

1 + z, and j(z) =

16 (z8 + 14 z4 + 1)3

z4 (z4 − 1)4.

A5 =⟨r , s

∣∣ s2 = r 3 = (s r)5 = 1⟩' PSL2(F4) ' PSL2(F5) are the rigid

rotations of the icosahedron and the dodecahedron, with

r(z) =

(ζ5 + ζ5

4) ζ5 − z(ζ5 + ζ5

4)z + ζ5

, s(z) =

(ζ5 + ζ5

4) − z(ζ5 + ζ5

4)z + 1

, j(z) =(z20 + 228 z15 + 494 z10 − 228 z5 + 1)3

z5 (z10 − 11 z5 − 1)5.



Moduli Spaces


Klein’s Theory of Covariants

Theorem (Felix Klein, 1884)

The rational function

j(z) =

−27

(8 z3 + 1

)3z3 (z3 − 1)3

for n = 3,

16

(z8 + 14 z4 + 1

)3z4 (z4 − 1)4

for n = 4,(z20 + 228 z15 + 494 z10 − 228 z5 + 1

)3z5 (z10 − 11 z5 − 1)5

for n = 5;

is invariant under the group G =⟨r , s

∣∣ s2 = r 3 = (s r)n = 1⟩

expressed interms of the generators

r(z) =

ζ3 z for n = 3,

ζ4 + z

ζ4 − zfor n = 4,(

ζ5 + ζ54)ζ5 − z(

ζ5 + ζ54)z + ζ5

for n = 5;

s(z) =

1 − z

2 z + 1for n = 3,

1 − z

1 + zfor n = 4,(

ζ5 + ζ54)− z(

ζ5 + ζ54)z + 1

for n = 5.

In particular, Gal(Q(ζn, z)/Q(ζn, j)

)' G ' PSL2(Z/nZ).



Moduli Spaces


Elliptic Curves Associated to Principal Polynomials

Theorem (Felix Klein, 1884; G, 1999)

Set n = 3, 4, 5. Let q(x) = xn + Axn−3 + · · ·+ B x + C be over K = Q(ζn)with splitting field L, and assume that Gal(L/K) ' PSL2(Z/nZ). Then thereexists j ∈ K such that LK = K(E [n]x) is the field generated by thex-coordinates of the n-torsion of an elliptic curve E with invariant j .

If we define the rational functions

λ(z) =

8 z3 + 1

z (z3 − 1)for n = 3,

(1 + ζ4)[z2 − (1 + ζ4) z − ζ4

] [z2 − (1− ζ4) z + ζ4

] [z2 + (1 + ζ4) z − ζ4

]z (z4 − 1)

for n = 4,

[z2 + 1

]2 [z2 + 2

(ζ5 + ζ5

4)z − 1

]2 [z2 + 2

(ζ5

2 + ζ53)z − 1

]2z(z10 − 11 z5 − 1

) + 3 for n = 5;

then we have the polynomials

q(x) =∏ν

[x −

1

λ(ζnν z

) ] =

x3 +1

jfor n = 3,

x4 +32

jx +

4

jfor n = 4,

x5 −40

jx2 −

5

jx −

1

jfor n = 5.



Moduli Spaces


Motivating Question

Let X be a compact Riemann surface. Fix a function φ : X → P1(C). For eachz in the inverse image of the thrice punctured sphere

φ−1

(P1(C)− {0, 1, ∞}

)⊆ X

form the elliptic curve

E : y 2 = x3 +3

φ(z)− 1x +

2

φ(z)− 1where j(E) =

1728

φ(z).

What are the properties of this elliptic curve?

Which types of functions φ : X → P1(C) are allowed?

If φ(z) has “lots of symmetries,” how does this translate into properties ofthe torsion elements K(E [n]x)?



Moduli Spaces


Belyı Maps

Theorem (Andre Weil, 1956; Gennadiı Vladimirovich Belyı, 1979)

A compact, connected Riemann surface X can be defined by a polynomialequation

∑i,j aij z

i w j = 0 where the coefficients aij are not transcendental if

and only if there exists a rational function φ : X → P1(C) which has at mostthree critical values

{0, 1, ∞

}.

“This discovery, which is technically so simple, made a very strong impressionon me, and it represents a decisive turning point in the course of myreflections, a shift in particular of my centre of interest in mathematics, whichsuddenly found itself strongly focused. I do not believe that a mathematicalfact has ever struck me quite so strongly as this one, nor had a comparablepsychological impact.

– Alexander Grothendieck, Esquisse d’un Programme (1984)

Definition

A rational function φ : X → P1(C) which has at most three critical values{0, 1, ∞

}is called a Belyı map.



Moduli Spaces


Dessins d’Enfant

Fix a Belyı map φ : X → P1(C). Denote the preimages

B = φ−1({0})

W = φ−1({1})

E = φ−1([0, 1]

)X

φ−−−−−→ P1(C)y y y y{black

vertices

} {white

vertices

} {edges

}R3

The bipartite graph Γφ =(V ,E

)with vertices V = B ∪W and edges E is

called Dessin d’Enfant. We embed the graph on X in 3-dimensions.

I do not believe that a mathematical fact has ever struck me quite so stronglyas this one, nor had a comparable psychological impact. This is surely becauseof the very familiar, non-technical nature of the objects considered, of whichany child’s drawing scrawled on a bit of paper (at least if the drawing is madewithout lifting the pencil) gives a perfectly explicit example. To such a dessinwe find associated subtle arithmetic invariants, which are completely turnedtopsy-turvy as soon as we add one more stroke.

– Alexander Grothendieck, Esquisse d’un Programme (1984)



Moduli Spaces


Must we work with

Platonic Solids?



Moduli Spaces

Rotation Group DnRotation Groups A4 and S4Rotation Group A5

Platonic Solids, Archimedean Solids, and Catalan Solids

Definition

A Platonic solid is a regular, convex polyhedron. They are named afterPlato (424 BC – 348 BC). Aside from the regular polygons, there are fivesuch solids.

An Archimedean solid is a convex polyhedron that has a similararrangement of nonintersecting regular convex polygons of two or moredifferent types arranged in the same way about each vertex with all sidesthe same length. Discovered by Johannes Kepler (1571 – 1630) in 1620,they are named after Archimedes (287 BC – 212 BC). Aside from theprisms and antiprisms, there are thirteen such solids.

A Catalan solid is a dual polyhedron to an Archimedean solid. They arenamed after Eugene Catalan (1814 – 1894) who discovered them in 1865.Aside from the bipyramids and trapezohedra, there are thirteen such solids.



Moduli Spaces


Platonic Solid Archimedean Solid Catalan Solid Rotation Group

Regular PolygonPrism Bipyramid

DnAntiprism Trapezohedron

Tetrahedron Truncated Tetrahedron Triakis Tetrahedron A4

Truncated Octahedron Tetrakis Hexahedron

Truncated Cube Triakis Octahedron

Octahedron Cuboctahedron Rhombic DodecahedronS4

Cube Truncated Cuboctahedron Disdyakis Dodecahedron

Rhombicuboctahedron Deltoidal Icositetrahedron

Snub Cube Pentagonal Icositetrahedron

Truncated Icosahedron Pentakis Dodecahedron

Truncated Dodecahedron Triakis Icosahedron

Icosahedron Icosidodecahedron Rhombic TriacontahedronA5

Dodecahedron Truncated Icosidodecahedron Disdyakis Triacontahedron

Rhombicosidodecahedron Deltoidal Hexecontahedron

Snub Dodecahedron Pentagonal Hexecontahedron



Moduli Spaces


Proposition (Wushi Goldring, 2012)

Let ψ(w) be a rational function. The composition ψ ◦ φ is also a Belyı map forevery Belyı map φ : X → P1(C) if and only if ψ is a Belyı map which sends theset{

0, 1, ∞}

to itself.

Proposition (Nicolas Magot and Alexander Zvonkin, 2000)

The following ψ are Belyı maps which send the set{

0, 1, ∞}

to itself.

ψ(w) =

−(w − 1)2/(4w) is a rectification,

(4w − 1)3/(27w) is a truncation,

1/w is a birectification,

(4− w)3/(27w 2) is a bitruncation,

4 (w 2 − w + 1)3/(27w 2 (w − 1)2

)is a rhombitruncation,

(w + 1)4/(16w (w − 1)2

)is a rhombification,

7496192 (w + θ)5

25 (3 + 8 θ)w(88w − (57 θ + 64)

)3 is a snubification,

where θ2 − (7/64) θ + 1 = 0.


http://en.wikipedia.org/wiki/Rectification_(geometry)

http://en.wikipedia.org/wiki/Truncation_(geometry)

http://en.wikipedia.org/wiki/Bitruncation_(geometry)


Moduli Spaces


Rotation Group Dn

Those semi-regular solids with group Dn =⟨r , s

∣∣ s2 = rn = (s r)2 = 1⟩

correspond to the vertices V = φ−1({0})

in terms of the Belyı maps

φ(z) =

(zn + 1)2

4 znfor the regular polygons,

(z2n + 14 zn + 1)3

108 zn (zn − 1)4for the prisms,

108 zn (zn − 1)4

(z2n + 14 zn + 1)3for the bipyramids,

(z2n + 10 zn − 2)4

16 zn (zn − 4)3 (2 zn + 1)3for the antiprisms,

16 zn (zn − 4)3 (2 zn + 1)3

(z2n + 10 zn − 2)4for the trapezohedra.



Moduli Spaces


Rotation Group Dn: Prism / Bipyramid

http://en.wikipedia.org/wiki/Prism_(geometry)

http://en.wikipedia.org/wiki/Bipyramid


http://en.wikipedia.org/wiki/Prism_(geometry)

http://en.wikipedia.org/wiki/Bipyramid


Moduli Spaces


Rotation Group Dn: Antiprism / Trapezohedron

http://en.wikipedia.org/wiki/Antiprism

http://en.wikipedia.org/wiki/Trapezohedra


http://en.wikipedia.org/wiki/Antiprism

http://en.wikipedia.org/wiki/Trapezohedra


Moduli Spaces


Rotation Groups A4, S4, and A5

Those semi-regular solids with group A4 =⟨r , s

∣∣ s2 = r 3 = (s r)3 = 1⟩

correspond to the tetrahedron.

φ(z) = −64 z3 (z3 − 1)3

(8 z3 + 1)3

Those semi-regular solids with group S4 =⟨r , s

∣∣ s2 = r 3 = (s r)4 = 1⟩

correspond to the the octahedron.

φ(z) =108 z4 (z4 − 1)4

(z8 + 14 z4 + 1)3

Those semi-regular solids with group A5 =⟨r , s

∣∣ s2 = r 3 = (s r)5 = 1⟩

correspond to the icosahedron.

φ(z) =1728 z5 (z10 − 11 z5 − 1)5

(z20 + 228 z15 + 494 z10 − 228 z5 + 1)3



Moduli Spaces


Rotation Group A4: Truncated Tetrahedron / Triakis Tetrahedron

http://en.wikipedia.org/wiki/Truncated_tetrahedron

http://en.wikipedia.org/wiki/Triakis_tetrahedron


http://en.wikipedia.org/wiki/Truncated_tetrahedron

http://en.wikipedia.org/wiki/Triakis_tetrahedron


Moduli Spaces


Rotation Group S4: Truncated Octahedron / Tetrakis Hexahedron

http://en.wikipedia.org/wiki/Truncated_octahedron

http://en.wikipedia.org/wiki/Tetrakis_hexahedron


http://en.wikipedia.org/wiki/Truncated_octahedron

http://en.wikipedia.org/wiki/Tetrakis_hexahedron


Moduli Spaces


Rotation Group S4: Truncated Cube / Triakis Octahedron

http://en.wikipedia.org/wiki/Truncated_cube

http://en.wikipedia.org/wiki/Triakis_octahedron


http://en.wikipedia.org/wiki/Truncated_cube

http://en.wikipedia.org/wiki/Triakis_octahedron


Moduli Spaces


Rotation Group S4: Cuboctahedron / Rhombic Dodecahedron

http://en.wikipedia.org/wiki/Cuboctahedron

http://en.wikipedia.org/wiki/Rhombic_dodecahedron


http://en.wikipedia.org/wiki/Cuboctahedron

http://en.wikipedia.org/wiki/Rhombic_dodecahedron


Moduli Spaces


Rotation Group S4: Truncated Cuboctahedron / Disdyakis Dodecahedron

http://en.wikipedia.org/wiki/Truncated_cuboctahedron

http://en.wikipedia.org/wiki/Disdyakis_dodecahedron


http://en.wikipedia.org/wiki/Truncated_cuboctahedron

http://en.wikipedia.org/wiki/Disdyakis_dodecahedron


Moduli Spaces


Rotation Group S4: Rhombicuboctahedron / Deltoidal Icositetrahedron

http://en.wikipedia.org/wiki/Rhombicuboctahedron

http://en.wikipedia.org/wiki/Deltoidal_icositetrahedron


http://en.wikipedia.org/wiki/Rhombicuboctahedron

http://en.wikipedia.org/wiki/Deltoidal_icositetrahedron


Moduli Spaces


Rotation Group S4: Snub Cube / Pentagonal Icositetrahedron

http://en.wikipedia.org/wiki/Snub_cube

http://en.wikipedia.org/wiki/Pentagonal_icositetrahedron


http://en.wikipedia.org/wiki/Snub_cube

http://en.wikipedia.org/wiki/Pentagonal_icositetrahedron


Moduli Spaces


Rotation Group A5: Truncated Icosahedron / Pentakis Dodecahedron

http://en.wikipedia.org/wiki/Truncated_icosahedron

http://en.wikipedia.org/wiki/Pentakis_dodecahedron


http://en.wikipedia.org/wiki/Truncated_icosahedron

http://en.wikipedia.org/wiki/Pentakis_dodecahedron


Moduli Spaces


Rotation Group A5: Truncated Dodecahedron / Triakis Icosahedron

http://en.wikipedia.org/wiki/Truncated_dodecahedron

http://en.wikipedia.org/wiki/Triakis_icosahedron


http://en.wikipedia.org/wiki/Truncated_dodecahedron

http://en.wikipedia.org/wiki/Triakis_icosahedron


Moduli Spaces


Rotation Group A5: Icosidodecahedron / Rhombic Triacontahedron

http://en.wikipedia.org/wiki/Icosidodecahedron

http://en.wikipedia.org/wiki/Rhombic_triacontahedron


http://en.wikipedia.org/wiki/Icosidodecahedron

http://en.wikipedia.org/wiki/Rhombic_triacontahedron


Moduli Spaces


Rotation Group A5: Truncated Icosidodecahedron / DisdyakisTriacontahedron

http://en.wikipedia.org/wiki/Truncated_icosidodecahedron

http://en.wikipedia.org/wiki/Disdyakis_triacontahedron


http://en.wikipedia.org/wiki/Truncated_icosidodecahedron

http://en.wikipedia.org/wiki/Disdyakis_triacontahedron


Moduli Spaces


Rotation Group A5: Rhombicosidodecahedron / Deltoidal Hexecontahedron

http://en.wikipedia.org/wiki/Rhombicosidodecahedron

http://en.wikipedia.org/wiki/Deltoidal_hexecontahedron


http://en.wikipedia.org/wiki/Rhombicosidodecahedron

http://en.wikipedia.org/wiki/Deltoidal_hexecontahedron


Moduli Spaces


Rotation Group A5: Snub Dodecahedron / Pentagonal Hexecontahedron

http://en.wikipedia.org/wiki/Snub_dodecahedron

http://en.wikipedia.org/wiki/Pentagonal_hexecontahedron


http://en.wikipedia.org/wiki/Snub_dodecahedron

http://en.wikipedia.org/wiki/Pentagonal_hexecontahedron


Moduli Spaces

X (3), X (4), and X (5)X0(2), X (2), and X (2, 4)Torsion on Elliptic Curves

Starting with the Belyı maps

φ(z) =

−64 z3 (z3 − 1)3

(8 z3 + 1)3for the tetrahedron i.e. A4,

108 z4 (z4 − 1)4

(z8 + 14 z4 + 1)3for the octahedron i.e. S4,

1728 z5 (z10 − 11 z5 − 1)5

(z20 + 228 z15 + 494 z10 − 228 z5 + 1)3for the icosahedron i.e. A5;

consider the composition ψ ◦ φ : P1(C)→ P1(C) in terms of the Belyı maps

ψ(w) =

−(w − 1)2/(4w) is a rectification,

(4w − 1)3/(27w) is a truncation,

1/w is a birectification,

(4− w)3/(27w 2) is a bitruncation,

4 (w 2 − w + 1)3/(27w 2 (w − 1)2

)is a rhombitruncation,

(w + 1)4/(16w (w − 1)2

)is a rhombification,

7496192 (w + θ)5

25 (3 + 8 θ)w(88w − (57 θ + 64)

)3 is a snubification,

where θ2 − (7/64) θ + 1 = 0.


http://en.wikipedia.org/wiki/Rectification_(geometry)

http://en.wikipedia.org/wiki/Truncation_(geometry)

http://en.wikipedia.org/wiki/Bitruncation_(geometry)


Moduli Spaces


Is There a

Moduli Space

Interpretation?



Moduli Spaces


Modular Curve Interpretation: Formulas for the Tetrahedron

X (3)z //

��

P1(C)

��

z(τ) =??

X0(3)ρ //

��

P1(C)

��

ρ(τ) = 729

[η(3τ)

η(τ)

]12

X (1)j // P1(C) j(τ) =

[1 + 240

∞∑n=1

σ3(n) qn

]3/[q∞∏n=1

(1− qn)24]

where q = e2πiτ . Since Aut(X (3)/X (1)

)' A4, we have the identities

ρ(τ) =27 z(τ)3

1− z(τ)3

j(τ) =

(ρ(τ) + 3

)3(ρ(τ) + 27

)ρ(τ)

= −27(8 z(τ)3 + 1

)3z(τ)3

(z(τ)3 − 1

)3Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids


Moduli Spaces


Modular Curve Interpretation: Formulas for the Octahedron

X (4)z //

��

P1(C)

��

z(τ) = 2

[η(τ)

η(2τ)

]2 [η(4τ)

η(2τ)

]4

X (2)λ //

��

P1(C)

��

λ(τ) =1

16

[η(τ)3

η(τ/2) η(2τ)2

]8

X (1)j // P1(C) j(τ) =

[1 + 240

∞∑n=1

σ3(n) qn

]3/[q∞∏n=1

(1− qn)24]


)' S4, we have the identities

λ(τ) =4 z(τ)2(

z(τ)2 + 1)2

j(z) =256

(λ(τ)2 − λ(τ) + 1

)3λ(τ)2

(λ(τ)− 1

)2 =16(z(τ)8 + 14 z(τ)4 + 1

)3z(τ)4

(z(τ)4 − 1



Moduli Spaces


Modular Curve Interpretation: Formulas for the Icosahedron

X (5)z //

��

P1(C)

��

z(τ) = − η(5τ)

η(τ)1/5

X0(5)µ //

��

P1(C)

��

µ(τ) = 125

[η(5τ)

η(τ)

]6X (1)

j // P1(C) j(τ) =

[1 + 240

∞∑n=1

σ3(n) qn

]3/[q∞∏n=1

(1− qn)24]


)' A5, we have the identities

µ(τ) =125 z(τ)5

z(τ)10 − 11 z(τ)5 − 1

j(τ) =

(µ(τ)2 + 10µ(τ) + 5

)3µ(τ)

=

(z(τ)20 + 228 z(τ)15 + 494 z(τ)10 − 228 z(τ)5 + 1

)3z(τ)5

(z(τ)10 − 11 z(τ)5 − 1



Moduli Spaces


Modular Curve Interpretation: Constructing Archimedean/Calatan Solids

X (2, 4)k //

��

P1(C)

��

k(τ) = 4

[η(τ) η(4τ)2

η(2τ)3

]4

X (2)λ //

��

P1(C)

��

λ(τ) = 16

[η(τ/2) η(2τ)2

η(τ)3

]8

X0(2)w //

��

P1(C)

��

w(τ) = −64

[η(2τ)

η(τ)

]24X (1)

J // P1(C) J(τ) =

[1 + 240

∞∑n=1

σ3(n) qn

]3/[1728 q

∞∏n=1

(1− qn)24]

w

(− 1

τ

)= −

(λ(τ)− 1

)24λ(τ)

w 7→ − (w − 1)2

4wrectification

J(τ) =

(4w(τ)− 1

)327w(τ)

w 7→ (4w − 1)3

27wtruncation

w

(− 1

2 τ

)=

1

w(τ)w 7→ 1

wbirectification



Moduli Spaces


Modular Curve Interpretation: Constructing Archimedean/Calatan Solids

X (2, 4)k //

��

P1(C)

��

k(τ) = 4

[η(τ) η(4τ)2

η(2τ)3

]4

X (2)λ //

��

P1(C)

��

λ(τ) = 16

[η(τ/2) η(2τ)2

η(τ)3

]8

X0(2)w //

��

P1(C)

��

w(τ) = −64

[η(2τ)

η(τ)

]24X (1)

J // P1(C) J(τ) =

[1 + 240

∞∑n=1

σ3(n) qn

]3/[1728 q

∞∏n=1

(1− qn)24]

J

(− 1

2 τ

)=

(4− w(τ)

)327w(τ)2

w 7→ (4− w)3

27w 2bitruncation

J(τ) =4(λ(τ)2 − λ(τ) + 1

)327λ(τ)2

(λ(τ)− 1

)2 w 7→ 4 (w 2 − w + 1)3

27w 2 (w − 1)2rhombitruncation

w

(τ

1− τ

)=

(k(τ) + 1

)416 k(τ)

(k(τ)− 1

)2 w 7→ (w + 1)4

16w (w − 1)2rhombification



Moduli Spaces


Big Question



Moduli Spaces


Motivating Question Revisited

Fix Belyı maps φ, ψ : P1(C)→ P1(C) as before. For each z in the inverseimage of the thrice punctured sphere(

ψ ◦ φ)−1(P1(C)− {0, 1, ∞}

)⊆ P1(C)

form the elliptic curve

E : y 2 = x3 +3(

ψ ◦ φ)(z)− 1

x +2(

ψ ◦ φ)(z)− 1

where j(E) =1728(

ψ ◦ φ)(z)

.

What are the properties of this elliptic curve?

PSL2(Z/nZ) ↪→ Gal(Q(ζn, z)/Q(ζn, j)

)for n = 3, 4, 5.

Since(ψ ◦ φ

)(z) has lots of symmetries, how does this translate into

properties of the torsion elements on E?

When ψ(w) = w , we see LK = Q(ζn, z) = K(E [n]x) contains the splittingfield L of many q(x) = xn + Axn−3 + · · ·+ B x + C over K = Q(ζn).



Moduli Spaces


Thank You!

Questions?


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