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Selections from the Arithmetic Geometry of Shimura Curves I: Modular Curves

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Selections from the Arithmetic Geometry of Shimura Curves I: Modular Curves SELECTIONS FROM THE ARITHMETIC GEOMETRY OF SHIMURA CURVES I: MODULAR CURVES PETE L. CLARK These are notes accompanying a sequence of three lectures given in the UGA num- ber theory seminar in August and September of 2006. I originally planned to give a single lecture on some Diophantine problems involving Atkin-Lehner twists of modular curves, but have been persuaded by Dino Lorenzini to expand it into three lectures and include Shimura curves as well. This ¯rst lecture is on (classical) modular curves. It is a rather \straight up" expository account of the subject, suitable for people who have heard about mod- ular curves before but seen little about them. After reviewing some truly classical1 results about modular curves over the complex numbers, we will discuss rational and integral canonical models, focusing in on the case of X0(N) for squarefree N. We hope to end this ¯rst lecture by revealing a (presumably) surprising connection between the geometry of modular curves and the arithmetic of quaternion algebras. 1. Uniformization of Riemann surfaces I will for the most part be making a bunch of interrelated statements of fact. Most of them are nontrivial and the reader should not be alarmed by their inability to supply a proof. Sometimes when some statement follows especially easily from an- other statement, I rapidly indicate a proof. Convention: All algebraic varieties we meet will be connected and nonsingular. In this section, all varieties will be de¯ned over C. The ¯rst fact that we will be taking for granted is that every (nonsingular!) algebaraic variety V=C gives rise to a complex manifold, whose set of points is V (C). Moreover, the connectedness of V in the algebraic sense { or, if you like, the connectedness of V (C) endowed with the Zariski topology - is equivalent to the connectedness of the associated complex manifold { or, if you like, the connectedness of V (C) endowed with the analytic topology (which is strictly ¯ner than the Zariski topology unless V (C) is ¯nite).2 Now suppose C is a complex algebraic curve, so that C(C) can be endowed with the structure of a Riemann surface (i.e., a one-dimensional complex manifold). On the other hand, not every Riemann surface arises this way. The precise condition for a Riemann surface to arise in this way { to be algebraic { is that it be the complement of a ¯nite set in a compact Riemann surface. In this case the com- pacti¯cation is well-determined up to isomorphism { in particular, its genus g(M) 1Classical does not imply trivial, of course, nor even devoid of contemporary research interest. 2Note that this is a special property of the complex numbers C. It does not even hold, e.g., for varieties over R. In fact the (¯nite) number of real-analytic components of a (Zariski-)connected variety V=R is an interesting invariant. 1 2 PETE L. CLARK is well-determined { as is the number p(M) of points required to compactify. (The corresponding algebraic curve is projective if p(M) = 0 and a±ne if p(M) > 0.) To say a bit more: given any connected Riemann surface M, the uniformization theorem asserts the following: (i) the universal covering M~ can be endowed with the structure of a Riemann surface in a unique way so that the covering map ¼ : M~ ! M is a local C-analytic isomorphism; and (ii) there are, up to equiva- lence, precisely three simply connected Riemann surfaces: the Riemann sphere P1; the Euclidean plane A1; and the upper half-plane H.3 P1 covers only itself.4 Given an unrami¯ed covering A1 ! M, we can view the deck transformation group ¤ as a lattice in A1 and hence M = A1=¤. Aside from the trivial case of ¤ = 0, we have the case in which ¤ =» Z and the quotient is isomorphic to Cn0, the multiplicative group; otherwise ¤ =» Z2 and M is an elliptic curve. In contrast to P1 and A1, the upper halfplane H is not itself algebraic, which makes it a much richer source of Riemann surfaces: indeed, by the process of elimination, 1 1 every Riemann surface which is not P ; A , Gm or an elliptic curve is uniformized by H: in particular every compact Riemann surface of genus at least 2. 2. Fuchsian groups In real life, most group actions have ¯xed points; in particular, it is important to expand our notation of \uniformized Riemann surface" to include (¯nite) ram- i¯cation. This goes as follows: ¯rst, the group of biholomorphic automorphisms of H is P SL2(R) = SL2(R)= § 1 acting by usual linear fractional transformations: az+b z 7! cz+d . 1 (Remarks: More generally we have GL2(C) acting on P (C) = C [ 1, GL2(R) § 1 1 + acting on H = P (C)nP (R), and GL2(R) , the matrices of positive determinant, acting on H. However, scalar matrices give rise to the trivial linear transformation, + and GL2(R) =scalars = P SL2(R).) By de¯nition, a Fuchsian group ¡ is a discrete subgroup of P SL2(R), so every Fuchsian group acts on H, and we may consider the map ¼¡ : H 7! ¡nH = Y (¡): It turns out that Y (¡) can be given the structure of a Riemann surface, uniquely speci¯ed by the condition that ¼¡ be a quotient map { that is, for any Riemann surface M, a function f : Y (¡) ! M is holomorphic i® its pullback f ± ¼¡ to H is holomorphic.5 Note well that this map is in general rami¯ed (over a discrete but 3The Riemann mapping theorem asserts that any simply connected proper domain in C is biholomorphic to H, so can be used in place of H. Occasionally it is also useful to work with the open unit disk. 4Exercise: Give as many di®erent proofs as you can, using (for instance) Euler characteristic, the Riemann-Hurwitz theorem, the Lefschetz Fixed point theorem, Luroth's Theorem::: 5This \categorical" perspective on quotients of manifolds with extra structure seems to be better known among European mathematicians, since it is used systematically in the Bourbaki books. SELECTIONS FROM THE ARITHMETIC GEOMETRY OF SHIMURA CURVES I: MODULAR CURVES3 at the moment possibly in¯nite set of points). Remark: In some sense this is a \lucky fact," turning on the especially simple local structure of rami¯ed ¯nite-to-one maps of Riemann surfaces (up to local bi- holomorphism, there are only the maps z 7! zn). If I am not mistaken, given a locally compact group G acting properly discontinuously on a locally compact space X, the only general assertion that can be made about the quotient GnX is that it is a Hausdor® space. In particular, the quotient of even a ¯nite group G of biholomorphic automorphisms of a higher-dimensional complex manifold can have singularities (\orbifold points"). What is the condition on ¡ such that Y (¡) be an algebraic Riemann surface? It is most easily phrased as follows: there exists on H a very nice hyperbolic6 metric dxdy ds2 = ; y2 in particular it is P SL2(R)-invariant. (After remarking that conformal automor- phisms of planar regions preserve angles and ds2, being a rescaling of the Euclidean metric, measures angles in the same way, what remains to be checked is that linear fractional transformations preserve hyperbolic length, which, if you like, is the ex- planation for why the factor of y2 has been inserted in the denominator.) We have the notion of a fundamental domain D for ¡ { this is the closure D ofS a connected open subset of H such that the ¡-translates of D tile H (i.e., H = γ2¡ γD, and distinct translates intersect only along @D) and of its volume v with respect to the hyperbolic metric.7 Theorem 1. For a Fuchsian group ¡, the quotient surface Y (¡) = ¡nH is algebraic i® ¡ has ¯nite hyperbolic volume. The special case in which Y (¡) is compact { i.e., ¡ has a compact fundamental domain { will be especially important in what follows. In older terminology, these are said to be Fuchsian groups of the ¯rst kind. Example: Take ¡(1) = P SL2(Z). Then, as is well-known, a fundamental domain 1 for ¡ can be obtained by taking j<(z)j · 2 , jzj ¸ 1, and the identi¯cations along the boundary are such that ¡(1)nH is, as a topological space, homeomorphic to A1. ¼ 1 Moreover, the volume is 3 < 1, so it is isomorphic to A as a Riemann surface (and not to the other, nonalgebraic Riemann surface to which it is homeomorphic, namely H). Note that the uniformization map ¼ : H! Y (1) is rami¯ed over two points of Y (1) (it is also, in a sense in which we are trying to avoid discussing in detail, in¯nitely rami¯ed over the missing point at 1), namely the distinguished points e2¼i=4 and e2¼i=6 of the fundamental region. These are, respectively, the unique ¯xed points in H of the elements S = ::: and T = :::. S and T are, up to conjugacy, the only nontrivial elements of ¡(1) of ¯nite order. It follows that if ¡ ½ ¡(1) is any subgroup containing no conjugacy class of S or T { and in 6We will ignore the temptation { quite strong in August 2006 { to digress on the general topic of geometric structures on manifolds. 7Since there are many di®erent fundamental domains for a given ¡, it formally speaking must be checked that the volume is independent of the chosen domain, but this is in fact rather easy. 4 PETE L. CLARK particular if ¡ is normal and contains neither S nor T { then the map ¼¡ : H! Y (¡) is the universal covering map.
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