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From Klein's Platonic Solids to Kepler's 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 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 What is a Platonic Solid? A regular, convex polyhedron is one of the collections of vertices 1 V = (z1 : z0) 2 P (C) δ(z1; z0) = 0 in terms of the homogeneous polynomials 8 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. 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 http://mathworld.wolfram.com/RegularPolygon.html http://en.wikipedia.org/wiki/Platonic_solids 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 Rigid Rotations of the 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) 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 Klein's Theory of Covariants Theorem (Felix Klein, 1884) The rational function 8 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 8 8 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). 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 Elliptic Curves Associated to Principal Polynomials 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 2 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 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 8 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) − f0; 1; 1g ⊆ 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 )? 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 Bely˘ıMaps Theorem (Andr´eWeil, 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; 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; 1 is called a Bely˘ımap.
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