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Canonical Models of Shimura Curves Canonical models of Shimura curves J.S. Milne April 4, 2003, v0.0 Abstract As an introduction to Shimura varieties, and, in particular, to Deligne’s Bourbaki and Corvallis talks (Deligne 1971, 1979), I explain the main ideas and results of the general theory of Shimura varieties in the context of Shimura curves. These notes had their origin in a two-hour lecture I gave on September 10, 2002. They are available at www.jmilne.org/math/. Please send corrections and comments to me at [email protected]. Contents Introduction . 3 1 Preliminaries 5 Algebraic varieties and their connected components . 5 Easy descent theory . 6 Adeles........................................` 8 2 Elliptic modular curves over C 12 ◦ The curve SΓ as a double coset space . 12 A finiteness statement . 14 The curve SK .................................... 14 Re-interpretation of X ............................... 17 Elliptic curves over C ................................ 18 Elliptic modular curves as parameter spaces over C . 20 Elliptic modular curves as moduli varieties over C . 23 3 Canonical models of elliptic modular curves 26 Statement of the main theorem . 26 Complex multiplication . 28 Special points . 30 Hard descent . 30 Existence of canonical models . 32 Definition of “canonical” . 32 1 CONTENTS 2 4 Automorphic vector bundles 33 5 Quaternionic Shimura curves 34 Quaternion algebras . 34 Quaternionic modular curves . 35 Quaternionic nonmodular curves . 36 6 Remarks on the general case 37 Deligne’s axioms . 37 Rough classification of Shimura varieties . 38 Main results on the existence of canonical models (post Shimura) . 38 CONTENTS 3 Introduction Let + X = {z ∈ C | =(z) > 0}. + Then SL2(R) acts transitively on X , a b az + b · z = , c d cz + d and the subgroup fixing i is the compact group a b 2 2 SO2 = a + b = 1 . −b a A congruence subgroup Γ of SL2(Z) is any subgroup containing the principal congruence subgroup of level N, a b a b 1 0 Γ(N) = ∈ SL2(Z) ≡ mod N . c d c d 0 1 Consider1 ◦ df + SΓ = Γ\X . Initially, this is a Riemann surface, but when a finite set of points (the “cusps”) is added, it becomes a compact Riemann surface, which is automatically2 a nonsingular projective ◦ 3 algebraic curve. Therefore, SΓ is an algebraic curve. ◦ ◦ The theorem I want to discuss is that SΓ has a canonical model CΓ over a certain number ◦ field FΓ. More precisely, there exists a curve CΓ over FΓ equipped with an isomorphism ◦ ◦ (CΓ)C → SΓ satisfying certain natural conditions sufficient to determine it uniquely. So one thing I’ll have to do is tell you how to attach a number field FΓ to a congruence subgroup. ◦ + Later, I’ll discuss a similar theorem for curves SΓ = Γ\X where Γ is again a congruence subgroup, but in a group different from SL2. Note that, in general, a variety over C will not have a model over a number field, and when it does, it will usually have many. For example, an elliptic curve E over C has a model over a number field if and only if its j-invariant j(E) is an algebraic number, and if Y 2Z = X3 + aXZ2 + bZ3 is one model of E over a number field k (meaning, a, b ∈ k), then Y 2Z = X3 + ac2XZ2 + bc3Z3 is a second which is isomorphic to the first only if c is 1Unfortunately, in his Bourbaki talk, Deligne writes this as X+/Γ. There used to be left-wingers (those who write the discrete group on the left) and right-wingers. Now there are only left-wingers — the right wingers either converted or . 2The functor C 7→ C(C) from nonsingular projective curves over C to compact Riemann surfaces is an equivalence of categories. For a discussion of this result, see my notes on modular forms, 7.3–7.7. 3The same is not true of X+, i.e., it is not possible to realize X+ (with its complex structure) as a Zariski- open subset of a nonsingular projective C curve over C — if you could, the complement of X+ in C(C) would be a finite set, and any bounded holomorphic function on X+ would extend to a bounded holomorphic C( ) z−i X+ z−i < 1 function on C , and so would be constant, but z+i is holomorphic on and z+i . This is one reason we work with Γ\X+ rather than X+ — as X+ is not an algebraic curve, it makes no sense to talk of it having a model over a number field. CONTENTS 4 a square in k. As another example, all the projective curves aX2 + bY 2 + cZ2 = 0 over Q become isomorphic to X2 + Y 2 + Z2 = 0 over C, but they fall into infinitely many distinct isomorphism classes over Q. A problem with the theorem as stated above is that the fields FΓ grow as Γ shrinks. One ◦ of Deligne’s innovations in his Bourbaki talk was to replace (in a systematic way) the SΓ with nonconnected curves which have canonical models over Q. Notations and terminology I use the language of algebraic varieties as, for example, in my course notes on algebraic geometry: the affine varieties over a field k are the ringed spaces Specm A with A a finitely generated k-algebra such that A ⊗ kal is reduced, and the varieties over k are the ringed spaces that are finite unions of open affine varieties satisfying a separatedness condition. Thus, a variety over k is essentially the same thing as a geometrically reduced separated scheme of finite type over k (not necessarily connected). For a variety V over k, k[V ] = Γ(V, OV ) is the ring of regular functions on V and, when V is irreducible, k(V ) is the field of rational functions on V . For simplicity, throughout the notes “variety” will mean “nonsingular variety”. With this convention, every connected variety is irreducible. Throughout, Qal is the algebraic closure of Q in C, and ι or z 7→ z¯ denotes complex conjugation on C. For a k-vector space V and a commutative k-algebra A, I often write V (A) for the A-module A ⊗k V . Given an equivalence relation, [*] denotes the equivalence class containing *. The notation X ≈ Y means that X and Y are isomorphic, whereas X ∼= Y means that they are canonically isomorphic or that there is a given (or unique) isomorphism.4 References In addition to the references listed at the end, I refer to the following of my course notes. GT Group Theory FT Fields and Galois Theory. AG Algebraic Geometry. ANT Algebraic Number Theory MF Modular Functions and Modular Forms EC Elliptic Curves. AV Abelian Varieties. CFT Class Field Theory. Prerequisites I assume some familiarity with the classical theory of elliptic modular curves as, for exam- ple, in the first four sections of MF. Acknowledgements I thank the following for providing corrections and comments for an earlier version of the notes: Brian Conrad. 4For example, if V is a finite-dimensional vector space over a field k, then V ≈ V ∨ and V =∼ V ∨∨. 1 PRELIMINARIES 5 1 Preliminaries Algebraic varieties and their connected components A variety V over a field k is said to be geometrically connected if Vkal is connected, in which case, VΩ is connected for every field Ω containing k (Hartshorne 1977, II, Exercise 3.15). We first examine zero-dimensional varieties. Over C, a zero-dimensional variety is nothing more than a finite set (finite disjoint union of copies A0). Over R, a connected zero-dimensional variety V is either geometrically connected (e.g., 0 ) or geometrically AR nonconnected (e.g., V : X2 + 1; subvariety of A1), in which case V (C) is a conjugate pair of complex points. Thus, one sees that to give a zero-dimensional variety over R is to give a finite set with an action of Gal(C/R). Similarly, a connected variety V over R may be geometrically connected, or it may decompose over C into a pair of conjugate varieties. Consider, for example, the following subvarieties of A2: L : Y + 1 is a geometrically connected line over R; L0 : Y 2 + 1 is connected over R, but over C it decomposes as the pair of conjugate lines Y = ±i. Note that R is algebraically closed5 in ∼ R[L] = R[X, Y ]/(Y + 1) = R[X] but not in 0 2 ∼ 2 ∼ R[L ] = R[X, Y ]/(Y + 1) = R[Y ]/(Y + 1) [X] = C[X]. PROPOSITION 1.1. A connected variety V over a field k is geometrically connected if and only if k is algebraically closed in k(V ). PROOF. This follows from the statement: let A be a finitely generated k-algebra such that al al A is an integral domain and A ⊗k k is reduced; then A ⊗ k is an integral domain if and only if k is algebraically closed in A (Zariski and Samuel 1958, III 15, Theorem 40). PROPOSITION 1.2. To give a zero-dimensional variety V over Q is to give (equivalently) (a) a finite set E plus, for each e ∈ E, a finite field extension Q(e) of Q, or 6 al 7 (b) a finite set S with a continuous (left) action of Σ =df Gal(Q /Q). PROOF. The underlying topological space V of a zero-dimensional variety (V, OV ) is finite and discrete, and for each e ∈ V , Γ(e, OV ) is a finite field extension of Q. al The set S in (b) is V (Q ) with the natural action of Σ. We can recover (V, OV ) from S as follows: let V be the set Σ\S of orbits endowed with the discrete topology, and, for al Σs e = Σs ∈ Σ\S, let Q(e) = (Q ) where Σs is the stabilizer of s in Σ; then, for U ⊂ V , Q Γ(U, OV ) = e∈U Q(e).
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