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Modular Curves Modular Curves David Loeer Notes by Florian Bouyer Copyright (C) Bouyer 2014. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license can be found at http://www.gnu.org/licenses/fdl.html Contents 0 Wae 2 0.1 Recap of modular forms . 2 1 Modular Curves as Riemann Surfaces 3 1.1 Modular curves as topological spaces . 3 1.2 Riemann surfaces: recap . 4 1.3 Genus, ramication, Riemann-Hurwitz . 4 1.4 Sheaves and Riemann-Roch . 6 1.5 The Katz sheaf . 7 2 Modular Curves as Algebraic Curves 9 2.1 Modular Curves over C ........................................... 9 2.2 Descending the base eld . 9 3 Modular Curves as Moduli Spaces 13 3.1 Lattices and Level Structures . 13 3.2 Moduli spaces and representable functors . 13 3.3 Elliptic curves over general base schemes . 14 3.4 Smoothness . 16 3.5 A complex - analytic digression . 17 3.6 Quotients and Y0(N) ............................................. 18 3.7 General Modular Curves . 19 3.8 General Level Structure . 20 4 Leftovers 21 4.1 Katz Modular Forms . 21 4.2 Cups and the Tate curve . 21 1 Why study modular curves? 1. They give more information on modular forms. 2. They give friendly examples of moduli spaces. 0 Wae Let . We have that the group acts on . Take of nite index. We dene H = fz 2 Cj im z > 0g SL2(R) H Γ < SL2(Z) Y (Γ) = ΓnH. We will equip this with various interesting structures. 0.1 Recap of modular forms Fix: with nite index (which we call the level) • Γ ≤ SL2(Z) • k 2 Z (which we call the weight) Then there exists a space of modular forms which are functions holomorphic, such that MK (Γ) f : H! C az + b a b f = (cz + d)kf(z) 8 2 Γ; cz + d c d and a growth condition on the boundary. There exists a subspace Sk(Γ) ⊂ MK (Γ) of cusp forms. Fact (Basic). Both and are nite dimensional over . Mk Sk C 1 h Any modular form has a q-expansion, let h be the least integer such that 2 Γ, then 0 1 X n 2πiz=h f(z) = anqh ; an 2 C; qh = e : n≥0 2 1 Modular Curves as Riemann Surfaces 1.1 Modular curves as topological spaces H has a topology (obviously) so Y (Γ) gets a quotient topology (i.e., strongest topology such that π : H! Γ(Y ) is continuous). Quotient topology can be pretty nasty (for example Q acting on R by translation) - quotient can even be the indiscrete topology. Proposition 1.1. For any there exists a neighbourhood such that if satises τ1; τ2 2 H U1 3 τ1;U2 3 τ2 γ 2 SL2(Z) γ(U1) \ U2 6= ;, then γ(τ1) = τ2. Proof. See Proposition 2.1.1 of Diamond and Shurman. We say that acts properly discontinuously on . SL2(Z) H Corollary 1.2. Y (Γ) is Hausdor Proof. Let P1 6= P2 be two points of Y (Γ). Choose, τ1; τ2 2 H be lifting of P1;P2 respectively. Let U1;U2 be neighbourhoods of τ1; τ2 as in Proposition 1.1. We claim that Vi = π(Ui) are open neighbourhoods of Pi such that V1 \ V2 = ;. −1 −1 0 0 Suppose V1 \ V2 6= ;, then π (V1) \ π (V2) 6= ;. Hence [γ2ΓγU1 \[γ02Γγ U2 6= ;. So there exists γ and γ 0 0 −1 0 −1 such that γU1 \ γ U2 6= ;. This gives (γ ) γU1 \ U2 6= ;, which by Proposition 1.1 means (γ ) γτ1 = τ2. This is a contradiction to the assumption that P1 and P2 are distinct point (hence τ1; τ2 are in dierent orbits) We are also interested in a slightly larger space X(Γ), which is a compactication of Y (Γ). As a set, we have X(Γ) = Y (Γ) [ C(Γ) | {z } "cusps" of Γ 1 C(Γ) = ΓnP (Q): Let H∗ = H[ P1(Q). Give H∗ a topology by extending the topology we have on H. If we dene neighbourhoods of 1 as fz : im z > Rg for any R 2 N. Then so that the topology make sense, we dened neighbourhood of x 2 Q as circles tangent to R at x: Actions on H∗ are still properly discontinuous so X(Γ) is Hausdor. Proposition 1.3. X(Γ) is compact. Proof. It suces to nd a compact subset of ∗ mapping surjectively to ). Let ∗ 1 H X(Γ D = f1g [ fz 2 H : 2 ≤ 1 . A standard fact is that ∗ contains a point of every orbit on ∗. So if are re z ≤ 2 ; jzj ≥ 1g D SL2(Z) H γ1; : : : ; γn coset representative for , then n ∗ surjects onto . It is easy to check that ∗ is compact, hence ΓnSL2(Z) [i=1γiD X(Γ) D we are done. 3 1.2 Riemann surfaces: recap Denition 1.4. A Riemann Surface consists of the following data: • A topological space X, (which is Hausdor and second - countable) A collection , where are opens forming a cover of , are opens in and • (Ui;Vi; φi)i2I Vi ⊂ X X Ui C φi : Ui ! Vi are homeomorphism, such that if , the map −1 −1 −1 is Vi \ Vj 6= ; φj ◦ φi : Ui \ φi (Vi \ Vj) ! Uj \ φj (Vi \ Vj) holomorphic. Roughly: A Riemann surface is the least amount of structure on X needed to make sense of a function X ! C being holomorphic. We will now show that Y (Γ) and X(Γ) have natural Riemann surface structures. Denition 1.5. We say is an elliptic point if for some (and hence any) lifting of , Stab , P 2 Y (Γ) τ 2 H P1 Γ(τ) 6= f1g where is the image of in PSL (that is ) Γ Γ 2(Z) Γ=Γ \ {±1g If is elliptic for , then it maps to an elliptic point of . One can see that there are only of these P Γ Y (SL2(Z)) 2 (the orbits of i and ρ = e2πi=3). If P is not elliptic or a cusp, we can easily nd a chart around P . Let τ be a lifting of P and apply Proposition 1.1 with τ1 = τ2 = τ. Let U = U1 \ U2, then U a neighbourhood of τ such that γU \ U = ; for any γ 6= 1 2 Γ. Let V be the image of U in Y (Γ), then φ = π=U is a homeomorphism U !∼ V . If P is a elliptic, need to be a bit cleverer. Proposition 1.1 gives us a U 3 τ such that U \ γU 6= ; if and only if Stab (which has order or ). Choose shifting to and to . Then Stab goes to γ 2 Γ(τ) 2 3 δ 2 SL2(C) τ 0 τ 1 Γ(τ) a nite cyclic group of Mobius transformations xing 0 and 1, hence multiplication by e2πi=n with n = 2 or 3. Dene the map Stab 0 by n, then this is a bijection. We dene the coordinate chart to be δΓδ−1 (0)nδU ! U z 7! z the map U 0 ! V as dened in the following map Stab 0 δΓδ−1 (0)nδU / U ∼ ∼ x V ⊂ Y (Γ) Lastly, if is a cusp, we argue similarly. Choose mapping to , Stab is a group of translation, P δ P 1 δΓδ−1 (1) and z 7! e2πiz=h gives a local coordinates. We have just proved that there exists a Riemann surface structure on Y (Γ) and X(Γ) such that π : H! Y (Γ) is holomorphic. (Clearly the unique such structure). 1.3 Genus, ramication, Riemann-Hurwitz Fact. Riemann surfaces are smooth orientable 2-manifolds, and there are not many of these. Compacts connected ones all look like doughnuts with g holes. Denition 1.6. We dene genus of a compact 2-manifold M, as the unique integer g = g(M) such that H1(M; Z) =∼ Z2g. The genus is closely related to the Euler Characteristic, P i i . If is as in the χ(M) = i≥0(−1) rk H (M; Z) M Fact, then H0 =∼ H2 =∼ Z and Hi = 0 for i ≥ 3, so χ(M) = 2 − 2g. Proposition 1.7. For , the space is isomorphic (as a Riemann surface, so in particular as Γ = SL2(Z) X(Γ) 2-manifold) to P1(C) =∼ S2. 4 Proof. The -invariant −1 is invariant and descend to a holomorphic map j j(z) = q + 744 + 196884q + ::: SL2(Z) 1 . It is bijective (counting zeroes using contour integration) so it has a holomorphic inverse. X(SL2 Z) ! P (C) Convention: Unless otherwise stated all Riemann surfaces are assumed to be connected. Recap: We want to nd for all . We know that has genus . We have that for all , there g(X(Γ)) Γ X(SL2(Z)) 0 Γ is a map X(Γ) X(SL2(Z)) Denition 1.8. 1. For f : X ! Y a non-constant morphism, P 2 X, the ramication degree, eP (f), is the unique integer e ≥ 1 e such that f looks like z 7! z locally. Note: points such that eP (f) > 1 are isolated. 2. If and are compact, the sum P (which is independent of ) is called the degree of . X Y P 2f −1(Q) ep(f) Q 2 Y f The degree of is PSL (which is the number of preimages of a generic point of X(Γ) ! X(SL2(Z)) [ 2(Z): Γ] X(SL2(Z)) Theorem 1.9 (Riemann - Hurwitz).
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