From Klein's Platonic Solids to Kepler's

Home , Archimedean solid, Bipyramid, Catalan solid, Hexecontahedron, Icositetrahedron, Pentakis dodecahedron

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 Examples due to Magot and Zvonkin 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 Bely˘ımaps. 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 Bely˘ımaps relate to moduli spaces of elliptic curves.

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Examples due to Magot and Zvonkin Moduli Spaces Outline of Talk

1 Dessin d’Enfants Platonic Solids Klein’s Theory of Covariants Bely˘ı’sTheorem

2 Examples due to Magot and Zvonkin 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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Platonic Solids Examples due to Magot and Zvonkin Klein’s Theory of Covariants Moduli Spaces Bely˘ı’sTheorem

Recap

A regular, convex polyhedron is one of the collections of vertices 1 V = (z1 : z0) ∈ P (C) δ(z1, z0) = 0

in terms of the homogeneous polynomials

 n n z1 + z0 for the regular polygon,   z z 3 − z 3 for the tetrahedron,  1 1 0   4 4 z1 z0 z1 − z0 for the octahedron,  δ(z , z ) = 8 4 4 8 1 0 z1 + 14 z1 z0 + z0 for the cube,   10 5 5 10 z1 z0 z1 − 11 z1 z0 − z0 for the icosahedron,   20 15 5 10 10  z1 + 228 z1 z0 + 494 z1 z0  for the dodecahedron.  5 15 20 − 228 z1 z0 + z0

2 We embed V ,→ S (R) into the unit sphere via stereographic projection.

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

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

1 1 Recall the action ◦ : PSL2(C) × P (C) → P (C). We seek Galois extensions Q(ζn, z)/Q(ζn, j) for rational j(z). n 2 n 2 Zn = r r = 1 and Dn = r, s s = r = (s r) = 1 are the rigid rotations of the regular convex polygons, with n 1 n 6912 z r(z) = ζn z, s(z) = , and j(z) = 1728 z or . z (zn + 1)2 2 3 3 A4 = r, s s = r = (s r) = 1 ' PSL2(F3) are the rigid rotations of the tetrahedron, with 1 − z 27 (8 z3 + 1)3 r(z) = ζ3 z, s(z) = , and j(z) = − . 2 z + 1 z3 (z3 − 1)3 2 3 4 S4 = r, s s = r = (s r) = 1 ' PGL2(F3) ' PSL2(Z/4 Z) are the rigid rotations of the octahedron and the cube, with 8 4 3 ζ4 + z 1 − z 16 (z + 14 z + 1) r(z) = , s(z) = , and j(z) = . 4 4 4 ζ4 − z 1 + z z (z − 1) 2 3 5 A5 = r, s s = r = (s r) = 1 ' PSL2(F4) ' PSL2(F5) are the rigid rotations of the icosahedron and the dodecahedron, with

4 4 20 15 10 5 3 ζ5 + ζ5 ζ5 − z ζ5 + ζ5 − z (z + 228 z + 494 z − 228 z + 1) r(z) = , s(z) = , j(z) = . 4 4 5 10 5 5 ζ5 + ζ5 z + ζ5 ζ5 + ζ5 z + 1 z (z − 11 z − 1)

Theorem (Felix Klein, 1884) The rational function  8 z3 + 13 −27 for n = 3,  z3 (z3 − 1)3   z8 + 14 z4 + 13 j(z) = 16 for n = 4, 4 4 4  z (z − 1)  20 15 10 5 3  z + 228 z + 494 z − 228 z + 1  for n = 5; z5 (z10 − 11 z5 − 1)5 2 3 n is invariant under the group G = r, s s = r = (s r) = 1 expressed in terms of the generators   1 − z ζ3 z for n = 3,  for n = 3,   2 z + 1    ζ4 + z  1 − z r(z) = for n = 4, s(z) = for n = 4, ζ4 − z 1 + z  4  4  ζ5 + ζ5 ζ5 − z  ζ5 + ζ5 − z  for n = 5;  for n = 5.  4  4 ζ5 + ζ5 z + ζ5 ζ5 + ζ5 z + 1

In particular, Gal Q(ζn, z)/Q(ζn, j) ' G ' PSL2(Z/n Z).

Theorem (Felix Klein, 1884; G, 1999)

n n−3 Set n = 3, 4, 5. Let q(x) = x + A x + ··· + 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 LK = 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  8 z3 + 1   for n = 3,  z (z3 − 1)    2 2 2  (1 + ζ4) z − (1 + ζ4) z − ζ4 z − (1 − ζ4) z + ζ4 z + (1 + ζ4) z − ζ4 λ(z) = for n = 4, z (z4 − 1)    h i2 h i2 h i2  z2 + 1 z2 + 2 ζ + ζ 4 z − 1 z2 + 2 ζ 2 + ζ 3 z − 1  5 5 5 5  + 3 for n = 5;  z z10 − 11 z5 − 1 then we have the polynomials  3 1 x + for n = 3,   j  " #  Y 1  32 4 q(x) = x − = x4 + x + for n = 4, ν j j ν λ ζn z     5 40 2 5 1 x − x − x − for n = 5. j j j Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Platonic Solids Examples due to Magot and Zvonkin Klein’s Theory of Covariants Moduli Spaces Bely˘ı’sTheorem

Motivating Question 1 Let X be a compact Riemann surface. Fix a function φ : X → P (C). For each z in the inverse image of the thrice punctured sphere −1 1 φ P (C) − {0, 1, ∞} ⊆ X form the elliptic curve 3 2 1728 E : y 2 = x 3 + x + where j(E) = . φ(z) − 1 φ(z) − 1 φ(z) What are the properties of this elliptic curve?

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

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

Theorem (AndréWeil, 1956; Gennadi˘ıVladimirovich Bely˘ı,1979) A compact, connected Riemann surface X can be defined by a polynomial P i j equation i,j aij z w = 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 Bely˘ımap.

1 Fix a Bely˘ımap φ : X → P (C). Denote the preimages

−1 −1 −1 φ 1 B = φ {0} W = φ {1} E = φ [0, 1] X −−−−−→ P (C)         y y y y black white edges 3 vertices vertices R 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 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)

Must we work with Platonic Solids?

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces 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 Eug`eneCatalan (1814 – 1894) who discovered them in 1865. Aside from the bipyramids and trapezohedra, there are thirteen such solids.

Platonic Solid Archimedean Solid Catalan Solid Rotation Group

Prism Bipyramid Regular Polygon Dn Antiprism Trapezohedron

Tetrahedron Truncated Tetrahedron Triakis Tetrahedron A4

Truncated Octahedron Tetrakis Hexahedron Truncated Cube Triakis Octahedron

Octahedron Cuboctahedron Rhombic Dodecahedron S4 Cube Truncated Cuboctahedron Disdyakis Dodecahedron Rhombicuboctahedron Deltoidal Icositetrahedron Snub Cube Pentagonal Icositetrahedron

Truncated Icosahedron Pentakis Dodecahedron Truncated Dodecahedron Triakis Icosahedron Icosahedron Icosidodecahedron Rhombic Triacontahedron A5 Dodecahedron Truncated Icosidodecahedron Disdyakis Triacontahedron Rhombicosidodecahedron Deltoidal Hexecontahedron

Snub Dodecahedron Pentagonal Hexecontahedron

Proposition (Wushi Goldring, 2012) Let ψ(w) be a rational function. The composition ψ ◦ φ is also a Bely˘ımap for 1 every Bely˘ımap φ : X → P (C) if and only if ψ is a Bely˘ımap which sends the set 0, 1, ∞ to itself.

Proposition (Nicolas Magot and Alexander Zvonkin, 2000) The following ψ are Bely˘ımaps 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) = 2 3 2 2 4 (w − w + 1) / 27 w (w − 1) is a rhombitruncation,  (w + 1)4/16 w (w − 1)2 is a rhombification,   7496192 (w + θ)5  is a snubification,  3  25 (3 + 8 θ) w 88 w − (57 θ + 64) where θ2 − (7/64) θ + 1 = 0.

2 n 2 Those semi-regular solids with group Dn = r, s s = r = (s r) = 1 correspond to the vertices V = φ−1{0} in terms of the Bely˘ımaps

 (z n + 1)2  for the regular polygons,  n  4 z    (z 2n + 14 z n + 1)3  for the prisms,  n n 4  108 z (z − 1)    108 z n (z n − 1)4 φ(z) = for the bipyramids, (z 2n + 14 z n + 1)3    2n n 4  (z + 10 z − 2)  for the antiprisms,  16 z n (z n − 4)3 (2 z n + 1)3    n n 3 n 3  16 z (z − 4) (2 z + 1)  for the trapezohedra.  (z 2n + 10 z n − 2)4

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

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

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

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

2 3 3 Those semi-regular solids with group A4 = r, s s = r = (s r) = 1 correspond to the tetrahedron.

64 z 3 (z 3 − 1)3 φ(z) = − (8 z 3 + 1)3

2 3 4 Those semi-regular solids with group S4 = r, s s = r = (s r) = 1 correspond to the the octahedron.

108 z 4 (z 4 − 1)4 φ(z) = (z 8 + 14 z 4 + 1)3

2 3 5 Those semi-regular solids with group A5 = r, s s = r = (s r) = 1 correspond to the icosahedron.

1728 z 5 (z 10 − 11 z 5 − 1)5 φ(z) = (z 20 + 228 z 15 + 494 z 10 − 228 z 5 + 1)3

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A4: Truncated Tetrahedron / Triakis Tetrahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Truncated Octahedron / Tetrakis Hexahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Truncated Cube / Triakis Octahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Cuboctahedron / Rhombic Dodecahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Truncated Cuboctahedron / Disdyakis Dodecahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Rhombicuboctahedron / Deltoidal Icositetrahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group S4: Snub Cube / Pentagonal Icositetrahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Truncated Icosahedron / Pentakis Dodecahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Truncated Dodecahedron / Triakis Icosahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Icosidodecahedron / Rhombic Triacontahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Truncated Icosidodecahedron / Disdyakis Triacontahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Rhombicosidodecahedron / Deltoidal Hexecontahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants Rotation Group Dn Examples due to Magot and Zvonkin Rotation Groups A4 and S4 Moduli Spaces Rotation Group A5 Rotation Group A5: Snub Dodecahedron / Pentagonal Hexecontahedron

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

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

Number Theory Seminar From Klein’s Platonic Solids to Kepler’s Archimedean Solids Dessin d’Enfants X (3), X (4), and X (5) Examples due to Magot and Zvonkin X0(2), X (2), and X (2, 4) Moduli Spaces Torsion on Elliptic Curves Starting with the Bely˘ımaps

 64 z 3 (z 3 − 1)3 − for the tetrahedron i.e. A ,  3 3 4  (8 z + 1)  108 z 4 (z 4 − 1)4 φ(z) = for the octahedron i.e. S , (z 8 + 14 z 4 + 1)3 4  5 10 5 5  1728 z (z − 11 z − 1)  for the icosahedron i.e. A5;  (z 20 + 228 z 15 + 494 z 10 − 228 z 5 + 1)3

1 1 consider the composition ψ ◦ φ : P (C) → P (C) in terms of the Bely˘ımaps  −(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) = 2 3 2 2 4 (w − w + 1) / 27 w (w − 1) is a rhombitruncation,  (w + 1)4/16 w (w − 1)2 is a rhombification,   7496192 (w + θ)5  is a snubification,  3  25 (3 + 8 θ) w 88 w − (57 θ + 64) where θ2 − (7/64) θ + 1 = 0.

Is There a Moduli Space Interpretation?

z 1 X (3) / P (C) z(τ) =??

12 ρ 1 η(3τ) X0(3) / ( ) ρ(τ) = 729 P C η(τ)

" ∞ #3 ," ∞ # j 1 X n Y n24 X (1) / P (C) j(τ) = 1 + 240 σ3(n) q q 1 − q n=1 n=1

2πiτ where q = e . Since Aut X (3)/X (1) ' A4, we have the identities 27 z(τ)3 ρ(τ) = 1 − z(τ)3

ρ(τ) + 33ρ(τ) + 27 27 8 z(τ)3 + 13 j(τ) = = − ρ(τ) z(τ)3 z(τ)3 − 13

2 4 z η(τ) η(4τ) X (4) / 1( ) z(τ) = 2 P C η(2τ) η(2τ)

3 8 λ 1 η(τ) X (2) / 1( ) λ(τ) = P C 16 η(τ/2) η(2τ)2

" ∞ #3 ," ∞ # j 1 X n Y n24 X (1) / P (C) j(τ) = 1 + 240 σ3(n) q q 1 − q n=1 n=1

2πiτ where q = e . Since Aut X (4)/X (1) ' S4, we have the identities 4 z(τ)2 λ(τ) = z(τ)2 + 12

256 λ(τ)2 − λ(τ) + 13 16 z(τ)8 + 14 z(τ)4 + 13 j(z) = = λ(τ)2 λ(τ) − 12 z(τ)4 z(τ)4 − 14

z η(5τ) X (5) / 1( ) z(τ) = − P C η(τ)1/5 6 µ 1 η(5τ) X0(5) / ( ) µ(τ) = 125 P C η(τ) " ∞ #3 ," ∞ # j 1 X n Y n24 X (1) / P (C) j(τ) = 1 + 240 σ3(n) q q 1 − q n=1 n=1

2πiτ where q = e . Since Aut X (5)/X (1) ' A5, we have the identities 125 z(τ)5 µ(τ) = z(τ)10 − 11 z(τ)5 − 1

µ(τ)2 + 10 µ(τ) + 53 j(τ) = µ(τ) z(τ)20 + 228 z(τ)15 + 494 z(τ)10 − 228 z(τ)5 + 13 = z(τ)5 z(τ)10 − 11 z(τ)5 − 15

2 4 k η(τ) η(4τ) X (2, 4) / 1( ) k(τ) = 4 P C η(2τ)3

2 8 λ η(τ/2) η(2τ) X (2) / 1( ) λ(τ) = 16 P C η(τ)3 24 w 1 η(2τ) X0(2) / ( ) w(τ) = −64 P C η(τ) " ∞ #3 ," ∞ # J 1 X n Y n24 X (1) / P (C) J(τ) = 1 + 240 σ3(n) q 1728 q 1 − q n=1 n=1

2 1 λ(τ) − 1 (w − 1)2 w − = − w 7→ − rectiﬁcation τ 4 λ(τ) 4 w 3 4 w(τ) − 1 (4 w − 1)3 J(τ) = w 7→ truncation 27 w(τ) 27 w 1 1 1 w − = w 7→ birectiﬁcation 2 τ w(τ) w

2 4 k η(τ) η(4τ) X (2, 4) / 1( ) k(τ) = 4 P C η(2τ)3

3 1 4 − w(τ) (4 − w)3 J − = w 7→ bitruncation 2 τ 27 w(τ)2 27 w 2 2 3 4 λ(τ) − λ(τ) + 1 4 (w 2 − w + 1)3 J(τ) = w 7→ rhombitruncation 27 λ(τ)2 λ(τ) − 12 27 w 2 (w − 1)2 4 τ k(τ) + 1 (w + 1)4 w = w 7→ rhombiﬁcation 1 − τ 16 k(τ) k(τ) − 12 16 w (w − 1)2

Big Question

Motivating Question Revisited 1 1 Fix Bely˘ımaps φ, ψ : P (C) → P (C) as before. For each z in the inverse image of the thrice punctured sphere −1 1 1 ψ ◦ φ P (C) − {0, 1, ∞} ⊆ P (C) form the elliptic curve 3 2 1728 E : y 2 = x 3 + x + where j(E) = . ψ ◦ φ(z) − 1 ψ ◦ φ(z) − 1 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 LK = Q(ζn, z) = K(E[n]x ) contains the splitting n n−3 ﬁeld L of many q(x) = x + A x + ··· + B x + C over K = Q(ζn).

Thank You!

Questions?

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