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Elliptic Curves Elliptic Curves JWR Friday November 30, 2001,7:40 AM This is an account of the talk of Cheol-Hyun Cho. The aim is to construct a \universal elliptic curve". Tong Hai Yang helped me with this - he told me about [1]. 1 Discrete Groups A good reference for the material in this section is [3]. 1. Throughout P = C denotes the Riemann sphere, H denotes the [ f1g upper half plane, C∗ denotes the multiplicative group of complex numbers, n n+1 and P = (C 0 )=C∗ denotes n dimensional complex projective space. n+1 n f g n For w C 0 let [w] := wC∗ denote the corresponding point of P . For 2 n f g A GLn+1(C) let MA denote the corresponding automorphism of projective space2 so that MA([w]) = [Aw]: Identify P1 and P via z = [z; 1] and = [1; 0] so that 1 az + b a b MA(z) = ;A = cz + d c d for A GL2(C). A transformation of form MA is called a M¨obiustrans- formation2 . 2. A matrix A SL2(R) I2 is called hyperbolic iff its eigenvalues are real and distinct,2 ellipticn {±iff itsg eigenvalues are distinct and not real (and therefore complex conjugate), and parabolic otherwise. It is easy to see that a nontrivial M¨obiustransformation has either exactly one or exactly 1 two fixed points; Hence a matrix A SL2(R) I2 is hyperbolic if and 2 n {± g only if the corresponding automorphism MA of P has two fixed points in R, elliptic if and only if MA has two fixed points one in H and the other in H, and parabolic if and only if it has exactly one fixed point. The fixed point− of a parabolic element lies in R . [ f1g 3. Let Γ SL2(R) be a subgroup. A point z H is called a regular point ⊂ 2 of Γ iff the isotropy group Γz is essentially trivial, i.e. Γz = Γ I2 .A \ {± g point z H is called an elliptic point of Γ iff it is a fixed point of MA some elliptic element2 A Γ. A point z R is called an cusp of Γ iff it is 2 2 [ f1g a fixed point of MA for some parabolic element A Γ. Denote by 2 X(Γ) := Γ(z): z H Cusp(Γ) ; Γ(z) := MA(z): A Γ f 2 [ g f 2 g the orbit space of Γ acting on the union of H with the set of cusps of Γ. For z H Cusp(Γ) let 2 [ Γz := A Γ: MA(z) = z f 2 g denote the stabilizer group of z. Points on the same orbit have conjugate (in Γ) stabilizer groups so orbits of Γ in H Cusp(Γ) may be classified as regular, elliptic, or cusp. It is easy to see that[ the stabilizer group SL2(R) := A SL2(R): MA( ) = 1 f 2 1 1g 1 h is the set of all real matrices A of form A = where h R. ± 0 1 2 Lemma 4. Let Γ SL2(R) be a discrete group. Then ⊂ (i) Let z0 H be an elliptic point of Γ. Then there is an element C SL2(C) 2 1 2 with MC (z0) = 0 and MC Γz M − a finite cyclic subgroup of the ◦ ◦ C stabilizer subgroup C∗ I2 of the origin in GL2(C). · (ii) Let x0 R is a cusp of Γ. Then is an element C SL2(R) 2 [ f1g 1 2 with MC (x0) = and MC Γz M − an infinite cyclic subgroup of 1 ◦ ◦ C SL2(R) . 1 5. Introduce a topology in H Cusp(Γ) by taking as a basis for the open [ sets the open sets of H together with all sets D z0 where z0 Cusp(Γ) and D is an open disk in H whose boundary is tangent[ f g to the R2 at [ f1g z0. (In case z0 = this means a set of form (z) > c.) Since M¨obius transformations map1 circles to circles it follows that= for every A Γ the map 2 MA : H Cusp(Γ) H Cusp(Γ) is a homeomorphism of in this topology. [ ! [ 2 Lemma 6. For every point z H Cusp(Γ) there is an open neighborhood U of z such that for A Γ we2 have[ 2 A Γz MA(U) U = : 2 () \ 6 ; Such an open set U is called an open slice centered at z. Lemma 7. Let Γ SL2(R) be a discrete group and z0 H Cusp(Γ). Then there is a continuous⊂ function ζ : U C defined in an2 open[ slice centered at ! z0 which is holomorphic on U z0 and such that for z; z0 U we have n f g 2 ζ(z0) = ζ(z) z0 Γ(z): () 2 (In case z0 H the function ζ is holomorphic on U since the singularity is removable.)2 8. A function ζ :U C as in Corollary 7 is called a local holomorphic ! invariant for Γ at z0. The injective map from U(Γ) := Γ(z): z U to C f 2 g induced by ζ is called a holomorphic coordinate for X(Γ) at Γ(z0). Theorem 9. Let Γ SL2(R) be a discrete group. Then the various holo- morphic coordinates for⊂ X(Γ) form an atlas. This atlas equips X(Γ) with the structure of a (Hausdorff) orbifold Riemann surface. Example 10. By the Uniformization Theorem a compact Riemann surface of genus greater than one is isomorphic to some X(Γ) where every element of Γ is hyperbolic. 1 h Example 11. Let Γ be the cyclic subgroup of SL (R) generated by 2 0 1 where h = 0. Then every point of H is regular, Cusp(Γ) = , and ζ(z) = e2πiz=h is6 a local holomorphic invariant for Γ. The holomorphicf1g coordinate induced by ζ on X(Γ) maps X(Γ) isomorphically to the unit disk. Remark 12. If Γ is a discrete subgroup of SL2(R) so is the group Γ0 generated by Γ and I2. Since as I2 acts trivially, the group Γ0 has the same orbits ± − as Γ so X(Γ0) = X(Γ). In particular, any discrete subgroup of SL2(R) has an orbit space identical to an X(Γ) as in Example 11. 1 3 Example 13. Let Γ = SL2(Z). Then Cusp(Γ) = Γ( ) = Q . There are two elliptic orbits Γ(i) and Γ(eπi=3). The stabilizer1 subgroups[ f1g at , i, and eπi=3 are generated by 1 1 1 0 1 1 1 A = ;B = ; AB = 0 1 1− 0 1− 0 respectively. The element A has infinite order, the element B has order four, and the element AB has order six. The transformation MA has infinite order, the transformation MB has order two, and the transformation MAB has order three. The function z e2πiz is a local invariant at , the function z ((z i)=(z + i))2 is a local7! invariant at i, and the function1 z πi=3 7! − πi=3 3 πi=3 7! ((z e )=(z e− )) is a local invariant at e . In Theorem 33 below construct− an isomorphism− from X(Γ) to projective space P; more precisely, to weighted projective space P(2; 4). 2 Lattices 14. A lattice is a subgroup of the additive group of C of form Λ = Z!1 + Z!2 where !1;!2 C are independent over R, i.e. one of !1=!2 and !2=!1 lies in the upper half2 plane H and the other in the lower half plane. Choose the indexing so τ := !1=!2 H. Then the automorphism z z=omega1 of C carries the lattice Λ to a2 lattice 7! Λτ := Z + Zτ where τ H. 2 Lemma 15. For τ; τ 0 H the following are equivalent: 2 (i) there exists A SL2(Z) with τ 0 = MA(τ); 2 (ii) there is an automorphism z αz + β of C with Λτ = αΛτ + β; 7! 0 Proof. Assume (i). Then Z(aτ + b) + Z(cτ + d) Z + Zτ = Λτ : ⊂ 4 The automorphism z z=(cτ +d) sends this aτ +b to 1 and 1 to τ 0 and hence 7! Λτ to Λτ . Since ad bc = 1 interchanging τ and τ 0 constructs the inverse 0 − homomorphism. Conversely assume (ii). Since 0 Λτ we have β Λτ so 2 2 0 Λτ = αΛτ 0 so ατ 0 and α generates Λτ . Hence there exist integers a; b; c; d with ad bc = 1, ατ 0 = aτ + b, α = cτ + d and hence τ 0 = (ατ 0)/α = MA(τ 0). − ± Since τ; τ 0 H we have ad bc = 1 (and not 1). 2 − − 16. Warning: Lemma 15 says when the lattices are isomorphic, not when they are identical. The condition that τ 0 Λτ implies that τ satisfies a quadratic equation with integer coefficients.2 There are only countably many Z such equations so for most τ we have Λτ 0 = Λτ for all A SL2( ) I . 6 2 Z Zn {± g It is not hard to see when Λτ 0 = Λτ . Two lattices Λ = !1 + !2 and Λ0 = Z!0 + Z!0 are equal if and only if there are integers a; b; c; d Z with 1 2 2 ad bc = 1 and !0 = a!1 + b!2, !0 = c!1 + d!2. Taking the cross product − ± 1 2 gives !0 !0 = !1 !2, i.e. the vectors !1 and !2 determine a parallelogram 1× 2 ± × with the same area as the one determined by !10 and !20 .
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