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

Transcription

Dessin d’Enfants
Examples due to Magot and Zvonkin
Moduli Spaces
From Klein’s Platonic Solids
to Kepler’s Archimedean Solids:
Elliptic Curves and Dessins d’Enfants
Part II
Edray Herber Goins
Department of Mathematics
Purdue University
September 7, 2012
Number Theory Seminar
From Klein’s Platonic Solids to Kepler’s Archimedean Solids
Dessin d’Enfants
Moduli Spaces
Abstract
In 1884, Felix Klein wrote his influential book, “Lectures on the
Icosahedron,” where he explained how to express the roots of any
quintic polynomial in terms of elliptic modular functions. His idea was
to relate rotations of the icosahedron with the automorphism group
of 5-torsion points on a suitable elliptic curve. In fact, he created a
theory which related rotations of each of the five regular solids (the
tetrahedron, cube, octahedron, icosahedron, and dodecahedron) with
the automorphism groups of 3-, 4-, and 5-torsion points.
Using modern language, the functions which relate the rotations with
elliptic curves are Belyı̆ maps. In 1984, Alexander Grothendieck
introduced the concept of a Dessin d’Enfant in order to understand
Galois groups via such maps. We will complete a circle of ideas by
reviewing 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) as
Dessins d’Enfant; and discussing how the corresponding Belyı̆ maps
relate to moduli spaces of elliptic curves.
Dessin d’Enfants
Moduli Spaces
Outline of Talk
1
Dessin d’Enfants
Platonic Solids
Klein’s Theory of Covariants
Belyı̆’s Theorem
2
Rotation Group Dn
Rotation Groups A4 and S4
Rotation Group A5
3
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Torsion on Elliptic Curves
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Recap
Dessin d’Enfants
Moduli Spaces
Platonic Solids
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
 n
z1 + z0 n






z1 z1 3 − z0 3







z z z 4 − z0 4

 1 0 1
δ(z1 , z0 ) = z1 8 + 14 z1 4 z0 4 + z0 8





10
5
5
10

z1 z0 z1 − 11 z1 z0 − z0





 z1 20 + 228 z1 15 z0 5 + 494 z1 10 z0 10



− 228 z1 5 z0 15 + z0 20
for the regular polygon,
for the tetrahedron,
for the octahedron,
for the cube,
for the icosahedron,
for the dodecahedron.
We embed V ,→ S 2 (R) into the unit sphere via stereographic projection.
Dessin d’Enfants
Moduli Spaces
Platonic Solids
http://mathworld.wolfram.com/RegularPolygon.html
http://en.wikipedia.org/wiki/Platonic_solids
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Rigid Rotations of the Platonic Solids
Recall the action ◦ : PSL2 (C) × P1 (C) → P1 (C). We seek Galois extensions
Q(ζn , z)/Q(ζn , j) for rational j(z).
Zn = r r n = 1 and Dn = r , s s 2 = r n = (s r )2 = 1 are the rigid
rotations of the regular convex polygons, with
r (z) = ζn z,
1
,
z
s(z) =
and
j(z) = 1728 z
n
or
6912 z n
.
(z n + 1)2
A4 = r , s s 2 = r 3 = (s r )3 = 1 ' PSL2 (F3 ) are the rigid rotations of
the tetrahedron, with
r (z) = ζ3 z,
s(z) =
1−z
,
2z + 1
and
j(z) = −
27 (8 z 3 + 1)3
.
z 3 (z 3 − 1)3
S4 = r , s s 2 = r 3 = (s r )4 = 1 ' PGL2 (F3 ) ' PSL2 (Z/4 Z) 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 (z 8 + 14 z 4 + 1)3
.
z 4 (z 4 − 1)4
A5 = r , s s 2 = 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
(z 20 + 228 z 15 + 494 z 10 − 228 z 5 + 1)3
j(z) =
z 5 (z 10 − 11 z 5 − 1)5
.
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Theorem (Felix Klein, 1884)
The rational function
3

8 z3 + 1

−27



z 3 (z 3 − 1)3 

3

z 8 + 14 z 4 + 1
j(z) = 16
4 (z 4 − 1)4

z


3


 z 20 + 228 z 15 + 494 z 10 − 228 z 5 + 1


z 5 (z 10 − 11 z 5 − 1)5
for n = 3,
for n = 4,
for n = 5;
is invariant under the group G = r , s s 2 = r 3 = (s r )n = 1 expressed in
terms of the generators

ζ3 z





 ζ4 + z
r (z) =
ζ4 − z 


ζ 5 + ζ5 4 ζ5 − z



ζ5 + ζ 5 4 z + ζ5
for n = 3,
for n = 4,
for n = 5;
 1−z





2z + 1

1−z
s(z) =
1+z




ζ 5 + ζ5 4 − z



4
ζ 5 + ζ5 z + 1
for n = 3,
for n = 4,
for n = 5.
In particular, Gal Q(ζn , z)/Q(ζn , j) ' G ' PSL2 (Z/n Z).
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Elliptic Curves Associated to Principal Polynomials
Theorem (Felix Klein, 1884; G, 1999)
Set n = 3, 4, 5. Let q(x) = x n + A x n−3 + · · · + B x + C be over K = Q(ζn )
with splitting field L, and assume that Gal(L/K ) ' PSL2 (Z/n Z). Then there
exists j ∈ K such that L K = K (E [n]x ) is the field generated by the
x-coordinates of the n-torsion of an elliptic curve E with invariant j.
If we define the rational functions
λ(z) =

8 z3 + 1





3
z

 (z − 1)




 (1 + ζ ) z 2 − (1 + ζ ) z − ζ z 2 − (1 − ζ ) z + ζ z 2 + (1 + ζ ) z − ζ 
4
4
4
4
4
4
4
z (z 4





h
i2 h
i2


2
2
4

 z +1
z + 2 ζ5 + ζ5
z −1




z z 10 − 11 z 5
− 1)
h
i2
z 2 + 2 ζ5 2 + ζ5 3 z − 1
+3
−1
for n = 3,
for n = 4,
for n = 5;
then we have the polynomials

1

x 3 +


j




"
#


Y
32
4
1
4
x+
q(x) =
x−
= x +

j
j
λ ζn ν z

ν





40 2
5
1


5
x −
x − x−
j
j
j
for n = 3,
for n = 4,
for n = 5.
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Motivating Question
Let X be a compact Riemann surface. Fix a function φ : X → P1 (C). For each
z in the inverse image of the thrice punctured sphere
φ−1 P1 (C) − {0, 1, ∞} ⊆ X
form the elliptic curve
E : y2 = x3 +
2
3
x+
φ(z) − 1
φ(z) − 1
where
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 of
the torsion elements K (E [n]x )?
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Belyı̆ Maps
Theorem (André Weil, 1956; Gennadiı̆ Vladimirovich Belyı̆, 1979)
A compact,
P connected Riemann surface X can be defined by a polynomial
equation i,j aij z i w j = 0 where the coefficients aij are not transcendental if
1
and only if there exists
a rational
function φ : X → P (C) which has at most
three critical values 0, 1, ∞ .
“This discovery, which is technically so simple, made a very strong impression
on me, and it represents a decisive turning point in the course of my
reflections, a shift in particular of my centre of interest in mathematics, which
suddenly found itself strongly focused. I do not believe that a mathematical
fact has ever struck me quite so strongly as this one, nor had a comparable
psychological impact.
– Alexander Grothendieck, Esquisse d’un Programme (1984)
Definition
1
A
rational function φ : X → P (C) which has at most three critical values
0, 1, ∞ is called a Belyı̆ map.
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Dessins d’Enfant
Fix a Belyı̆ map φ : X → P1 (C). Denote the preimages
φ
W = φ−1 {1}
E = φ−1 [0, 1]
X −
−−−−
→ P1 (C)






y
y
y
white
edges
R3
vertices
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.
B = φ−1 {0}


y
black
vertices
I do not believe that a mathematical fact has ever struck me quite so strongly
as this one, nor had a comparable psychological impact. This is surely because
of the very familiar, non-technical nature of the objects considered, of which
any child’s drawing scrawled on a bit of paper (at least if the drawing is made
without lifting the pencil) gives a perfectly explicit example. To such a dessin
we find associated subtle arithmetic invariants, which are completely turned
topsy-turvy as soon as we add one more stroke.
– Alexander Grothendieck, Esquisse d’un Programme (1984)
Dessin d’Enfants
Moduli Spaces
Platonic Solids
Must we work with
Platonic Solids?
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Platonic Solids, Archimedean Solids, and Catalan Solids
Definition
A Platonic solid is a regular, convex polyhedron. They are named after
Plato (424 BC – 348 BC). Aside from the regular polygons, there are five
such solids.
An Archimedean solid is a convex polyhedron that has a similar
arrangement of nonintersecting regular convex polygons of two or more
different types arranged in the same way about each vertex with all sides
the same length. Discovered by Johannes Kepler (1571 – 1630) in 1620,
they are named after Archimedes (287 BC – 212 BC). Aside from the
prisms and antiprisms, there are thirteen such solids.
A Catalan solid is a dual polyhedron to an Archimedean solid. They are
named after Eugène Catalan (1814 – 1894) who discovered them in 1865.
Aside from the bipyramids and trapezohedra, there are thirteen such solids.
Dessin d’Enfants
Moduli Spaces
Platonic Solid
Regular Polygon
Tetrahedron
Archimedean Solid
Rotation Group Dn
Rotation Group A5
Catalan Solid
Prism
Bipyramid
Antiprism
Trapezohedron
Truncated Tetrahedron
Triakis Tetrahedron
Truncated Octahedron
Tetrakis Hexahedron
Truncated Cube
Triakis Octahedron
Octahedron
Cuboctahedron
Rhombic Dodecahedron
Cube
Truncated Cuboctahedron
Disdyakis Dodecahedron
Rhombicuboctahedron
Deltoidal Icositetrahedron
Snub Cube
Pentagonal Icositetrahedron
Truncated Icosahedron
Pentakis Dodecahedron
Truncated Dodecahedron
Triakis Icosahedron
Icosahedron
Icosidodecahedron
Rhombic Triacontahedron
Dodecahedron
Truncated Icosidodecahedron
Disdyakis Triacontahedron
Rhombicosidodecahedron
Deltoidal Hexecontahedron
Snub Dodecahedron
Pentagonal Hexecontahedron
Rotation Group
Dn
A4
S4
A5
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Proposition (Wushi Goldring, 2012)
Let ψ(w ) be a rational function. The composition ψ ◦ φ is also a Belyı̆ map for
1
every
Belyı̆ map
φ : X → P (C) if and only if ψ is a Belyı̆ map which sends the
set 0, 1, ∞ to itself.
Proposition (Nicolas Magot and Alexander Zvonkin, 2000)
The following ψ are Belyı̆ maps which send the set 0, 1, ∞ to itself.


−(w − 1)2 /(4 w )
is a rectification,



3


(4 w − 1) /(27 w )
is a truncation,




1/w
is
a birectification,




(4 − w )3 /(27 w 2 )
is a bitruncation,
ψ(w ) =

4 (w 2 − w + 1)3 / 27 w 2 (w − 1)2
is a rhombitruncation,




4
2

(w
+
1)
/
16
w
(w
−
1)
is a rhombification,




7496192 (w + θ)5



3 is a snubification,

25 (3 + 8 θ) w 88 w − (57 θ + 64)
where θ2 − (7/64) θ + 1 = 0.
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group Dn
Those semi-regular solids with group Dn = r , s s 2 = r n = (s r )2 = 1
−1
correspond to the vertices V = φ
{0} in terms of the Belyı̆ maps
 n
(z + 1)2


for the regular polygons,

 4 zn








(z 2n + 14 z n + 1)3


for the prisms,

 108 z n (z n − 1)4







108 z n (z n − 1)4
φ(z) =
for the bipyramids,
2n + 14 z n + 1)3

(z







(z 2n + 10 z n − 2)4



for the antiprisms,


16 z n (z n − 4)3 (2 z n + 1)3







16 z n (z n − 4)3 (2 z n + 1)3



for the trapezohedra.
(z 2n + 10 z n − 2)4
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group Dn : Prism / Bipyramid
http://en.wikipedia.org/wiki/Prism_(geometry)
http://en.wikipedia.org/wiki/Bipyramid
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group Dn : Antiprism / Trapezohedron
http://en.wikipedia.org/wiki/Antiprism
http://en.wikipedia.org/wiki/Trapezohedra
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Groups A4 , S4 , and A5
Those semi-regular solids with group A4 = r , s s 2 = r 3 = (s r )3 = 1
correspond to the tetrahedron.
φ(z) = −
64 z 3 (z 3 − 1)3
(8 z 3 + 1)3
Those semi-regular solids with group S4 = r , s s 2 = r 3 = (s r )4 = 1
correspond to the the octahedron.
φ(z) =
108 z 4 (z 4 − 1)4
(z 8 + 14 z 4 + 1)3
Those semi-regular solids with group A5 = r , s s 2 = r 3 = (s r )5 = 1
correspond to the icosahedron.
φ(z) =
(z 20
1728 z 5 (z 10 − 11 z 5 − 1)5
+ 228 z 15 + 494 z 10 − 228 z 5 + 1)3
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A4 : Truncated Tetrahedron / Triakis Tetrahedron
http://en.wikipedia.org/wiki/Truncated_tetrahedron
http://en.wikipedia.org/wiki/Triakis_tetrahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Truncated Octahedron / Tetrakis Hexahedron
http://en.wikipedia.org/wiki/Truncated_octahedron
http://en.wikipedia.org/wiki/Tetrakis_hexahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Truncated Cube / Triakis Octahedron
http://en.wikipedia.org/wiki/Truncated_cube
http://en.wikipedia.org/wiki/Triakis_octahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Cuboctahedron / Rhombic Dodecahedron
http://en.wikipedia.org/wiki/Cuboctahedron
http://en.wikipedia.org/wiki/Rhombic_dodecahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Truncated Cuboctahedron / Disdyakis Dodecahedron
http://en.wikipedia.org/wiki/Truncated_cuboctahedron
http://en.wikipedia.org/wiki/Disdyakis_dodecahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Rhombicuboctahedron / Deltoidal Icositetrahedron
http://en.wikipedia.org/wiki/Rhombicuboctahedron
http://en.wikipedia.org/wiki/Deltoidal_icositetrahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group S4 : Snub Cube / Pentagonal Icositetrahedron
http://en.wikipedia.org/wiki/Snub_cube
http://en.wikipedia.org/wiki/Pentagonal_icositetrahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Truncated Icosahedron / Pentakis Dodecahedron
http://en.wikipedia.org/wiki/Truncated_icosahedron
http://en.wikipedia.org/wiki/Pentakis_dodecahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Truncated Dodecahedron / Triakis Icosahedron
http://en.wikipedia.org/wiki/Truncated_dodecahedron
http://en.wikipedia.org/wiki/Triakis_icosahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Icosidodecahedron / Rhombic Triacontahedron
http://en.wikipedia.org/wiki/Icosidodecahedron
http://en.wikipedia.org/wiki/Rhombic_triacontahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Truncated Icosidodecahedron / Disdyakis
Triacontahedron
http://en.wikipedia.org/wiki/Truncated_icosidodecahedron
http://en.wikipedia.org/wiki/Disdyakis_triacontahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Rhombicosidodecahedron / Deltoidal Hexecontahedron
http://en.wikipedia.org/wiki/Rhombicosidodecahedron
http://en.wikipedia.org/wiki/Deltoidal_hexecontahedron
Dessin d’Enfants
Moduli Spaces
Rotation Group Dn
Rotation Group A5
Rotation Group A5 : Snub Dodecahedron / Pentagonal Hexecontahedron
http://en.wikipedia.org/wiki/Snub_dodecahedron
http://en.wikipedia.org/wiki/Pentagonal_hexecontahedron
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Starting with the Belyı̆ maps

64 z 3 (z 3 − 1)3


−



(8 z 3 + 1)3


108 z 4 (z 4 − 1)4
φ(z) =

(z 8 + 14 z 4 + 1)3



1728 z 5 (z 10 − 11 z 5 − 1)5


 20
(z + 228 z 15 + 494 z 10 − 228 z 5 + 1)3
consider the composition ψ ◦ φ : P1 (C) → P1 (C) in

2

−(w − 1) /(4 w )


3


(4 w − 1) /(27 w )




1/w



(4 − w )3 /(27 w 2 )
ψ(w ) =
4 (w 2 − w + 1)3 / 27 w 2 (w − 1)2





(w + 1)4 / 16 w (w − 1)2




7496192 (w + θ)5



3

25 (3 + 8 θ) w 88 w − (57 θ + 64)
for the tetrahedron i.e. A4 ,
for the octahedron i.e. S4 ,
for the icosahedron i.e. A5 ;
terms of the Belyı̆ maps
is
is
is
is
is
is
a
a
a
a
a
a
rectification,
truncation,
birectification,
bitruncation,
rhombitruncation,
rhombification,
is a snubification,
where θ2 − (7/64) θ + 1 = 0.
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Is There a
Moduli Space
Interpretation?
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Modular Curve Interpretation: Formulas for the Tetrahedron
X (3)
X0 (3)
X (1)
z
/ P1 (C)
ρ
/ P1 (C)
j
/ P1 (C)
z(τ ) =??
ρ(τ ) = 729
"
j(τ ) = 1 + 240
∞
X
n=1
η(3τ )
η(τ )
12
#3 ,"
σ3 (n) q
n
q
∞
Y
#
1−q
n 24
n=1
where q = e 2πiτ . Since Aut X (3)/X (1) ' A4 , we have the identities
ρ(τ ) =
j(τ ) =
27 z(τ )3
1 − z(τ )3
3
3
ρ(τ ) + 3 ρ(τ ) + 27
27 8 z(τ )3 + 1
=−
3
ρ(τ )
z(τ )3 z(τ )3 − 1
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Modular Curve Interpretation: Formulas for the Octahedron
X (4)
X (2)
X (1)
z
λ
j
/ P1 (C)
z(τ ) = 2
/ P1 (C)
λ(τ ) =
"
/ P (C)
1
j(τ ) = 1 + 240
η(τ )
η(2τ )
2 η(4τ )
η(2τ )
4
8
η(τ )3
1
16 η(τ /2) η(2τ )2
∞
X
n=1
#3 ,"
σ3 (n) q
n
q
∞
Y
#
1−q
n 24
n=1
where q = e 2πiτ . Since Aut X (4)/X (1) ' S4 , we have the identities
λ(τ ) =
j(z) =
4 z(τ )2
z(τ )2 + 1
2
3
3
256 λ(τ )2 − λ(τ ) + 1
16 z(τ )8 + 14 z(τ )4 + 1
=
2
4
λ(τ )2 λ(τ ) − 1
z(τ )4 z(τ )4 − 1
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Modular Curve Interpretation: Formulas for the Icosahedron
X (5)
X0 (5)
X (1)
z
/ P1 (C)
µ
/ P1 (C)
j
/ P1 (C)
η(5τ )
η(τ )1/5
6
η(5τ )
µ(τ ) = 125
η(τ )
#
"
#
3 ," ∞
∞
X
Y
n 24
n
1−q
j(τ ) = 1 + 240
σ3 (n) q
q
z(τ ) = −
n=1
n=1
where q = e 2πiτ . Since Aut X (5)/X (1) ' A5 , we have the identities
125 z(τ )5
− 11 z(τ )5 − 1
3
µ(τ )2 + 10 µ(τ ) + 5
j(τ ) =
µ(τ )
µ(τ ) =
=
z(τ )10
z(τ )20 + 228 z(τ )15 + 494 z(τ )10 − 228 z(τ )5 + 1
5
z(τ )5 z(τ )10 − 11 z(τ )5 − 1
3
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Modular Curve Interpretation: Constructing Archimedean/Calatan Solids
X (2, 4)
X (2)
X0 (2)
X (1)
k
λ
w
J
/ P1 (C)
k(τ ) = 4
/ P1 (C)
η(τ ) η(4τ )2
η(2τ )3
8
η(τ /2) η(2τ )2
λ(τ ) = 16
η(τ )3
24
η(2τ )
w (τ ) = −64
η(τ )
"
#3 ,"
#
∞
∞
X
Y
n 24
n
J(τ ) = 1 + 240
1728 q
1−q
σ3 (n) q
/ P1 (C)
/ P1 (C)
n=1
w
−
1
τ
4
=−
λ(τ ) − 1
4 λ(τ )
2
4 w (τ ) − 1
J(τ ) =
27 w (τ )
1
1
=
w −
2τ
w (τ )
w 7→ −
3
n=1
(w − 1)2
4w
(4 w − 1)3
27 w
1
w→
7
w
w 7→
rectification
truncation
birectification
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Modular Curve Interpretation: Constructing Archimedean/Calatan Solids
X (2, 4)
X (2)
X0 (2)
X (1)
k
λ
w
J
/ P1 (C)
/ P1 (C)
/ P1 (C)
/ P1 (C)
k(τ ) = 4
η(τ ) η(4τ )2
η(2τ )3
8
η(τ /2) η(2τ )2
η(τ )3
24
η(2τ )
w (τ ) = −64
η(τ )
#3 ,"
#
"
∞
∞
X
Y
n
n 24
1728 q
J(τ ) = 1 + 240
σ3 (n) q
1−q
λ(τ ) = 16
n=1
J
4
3
4 − w (τ )
1
−
=
2τ
27 w (τ )2
3
4 λ(τ )2 − λ(τ ) + 1
2
27 λ(τ )2 λ(τ ) − 1
4
k(τ ) + 1
τ
w
=
2
1−τ
16 k(τ ) k(τ ) − 1
J(τ ) =
n=1
w 7→
(4 − w )3
27 w 2
bitruncation
w 7→
4 (w 2 − w + 1)3
27 w 2 (w − 1)2
rhombitruncation
w 7→
(w + 1)4
16 w (w − 1)2
rhombification
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Big Question
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Motivating Question Revisited
Fix Belyı̆ maps φ, ψ : P1 (C) → P1 (C) as before. For each z in the inverse
image of the thrice punctured sphere
−1 1
ψ◦φ
P (C) − {0, 1, ∞} ⊆ P1 (C)
form the elliptic curve
E : y2 = x3 +
2
3
x+
ψ ◦ φ (z) − 1
ψ ◦ φ (z) − 1
where
j(E ) =
1728
.
ψ ◦ φ (z)
What are the properties of this elliptic curve?
PSL2 (Z/n Z) ,→ 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 L K = Q(ζn , z) = K (E [n]x ) contains the splitting
field L of many q(x) = x n + A x n−3 + · · · + B x + C over K = Q(ζn ).
Dessin d’Enfants
Moduli Spaces
X (3), X (4), and X (5)
X0 (2), X (2), and X (2, 4)
Thank You!
Questions?

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

Transcription

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Robert Gene Klein was born on February 10, 1928`to Bert and Kate