CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES

Transcription

1 CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES FRANK GOUNELAS 1. Class Field Theory We ll begin by motivating some of the constructions of the CM (complex multiplication) theory for elliptic curves by mentioning the key results in class field theory. We know that a prime ideal in a number field K ramifies if and only if it divides the discriminant of that number field. Minkowski s bound thus implies that there are no unramified extensions of the rationals. Recall that the Hilbert class field of K is the maximal abelian unramified extension of K (in the algebraic closure K). The degree of the extension of the Hilbert class field over K is the class number and the Galois group of this extension is canonically isomorphic to the ideal class group of K. Hence if a number field has non-trivial class number, it has unramified extensions (namely the Hilbert class field) Local class field theory. The role of class field theory is to study all abelian extensions (the Galois group of the extension is abelian) of global and local fields. We ll state the main theorems of class field theory in the local case and use this to build the case of global fields. Recall that by a non-archimedean local field we mean a finite extension of either Q p for some prime number p or of F q ((T)) for q a prime power. We will assume that a local field is non-archimedean. By an unramified extension L/K we mean one such that Gal(L/K) = Gal(l/k) where l/k the residue field extension is separable). Roughly, the main idea is that for L/K an abelian extension we want to compute Gal(L/K) in such a way that L Gal(L/K) = K is more explicitly described. In modern language, this works by making calculations with the Tate cohomology groups Ĥi (Gal(L/K),L ), for example we have Ĥ2 (Gal(L/K),L ) = 1 [L:K] Z/Z. All these finite extensions can be assembled to K ab /K and one works hard to prove that Br(K) = Ĥ2 (Gal(K/K),K ) = Ĥ 2 (Gal(K nr /K),K nr, ) = Ĥ2 (Gal(K ab /K),K ab, ) = Q/Z. The main theorems in local class field theory are assembled in the following statement. Theorem 1.1. (Local class field theory)(see [Mil08], [Ser79]) Let K be a local field and let K ab /K be the maximal abelian extension of K. For finite abelian extensions L/K the relative local Artin maps φ L/K : K /Nm L/K (L ) Gal(L/K) Date: November 8,

2 2 FRANK GOUNELAS are isomorphisms and there is an order preserving bijection between finite abelian extensions L/K and open subgroups of finite index in K via L Nm L/K (L ). The relative local Artin maps form an inverse system which can be assembled to the local Artin map φ K : K Gal(K ab /K). The map φ L/K is really the inverse of a coboundary map composed with a cup product in Tate cohomology and more specifically Tate s Theorem φ 1 L/K : Gal(L/K) = Ĥ 2 (Gal(L/K), Z) Ĥ0 (Gal(L/K),L ) = K /Nm L/K (L ) In the special case where L/K is unramified, the Galois group is cyclic and the generator is called the Frobenius element Frob Gal(l/k). In this case φ L/K (a) = Frob ord K(a). After choosing a uniformising element π of K, we can write K = U K π bz for U K the units in K (there are structure theorems in place for determining this group) and Ẑ = p Z p. Corollary 1.2. (Local Kronecker-Weber Theorem)([Mil08] I.1.15) Let K ab /K be the maximal abelian extension of a local field K. Then K ab = K π K nr where K nr /K is the maximal unramified extension of K in K ab (i.e. the subfield fixed by φ K (U K )) and K π is the subfield of K ab fixed by φ K (π). In the case of K = Q p we can say exactly what these fields are. Namely, we obtain a decomposition Q ab p = Q[ζ p n] Q[ζ m ] n (m,p)= Global class field theory. From now on, let K be a global field (i.e. a number field or a finite extension of F q (T) for q a prime power). We will stick to considering number fields as they are more intuitive but all the results hold in both cases. We will outline the idèlic approach to global class field theory as it better exhibits the local-to-global philosophy which underlies these ideas. Define the group of adèles of K as A K = v K v where the product ranges over all (infinite and finite) places of K and all but finitely many of the terms are in O Kv (so that the end space is locally compact). Similarly define the idèles as I K = A K = v K v with the same conditions on the product and finally let C K = I K /K denote the idèle class group. Join all finite extensions of K together and get I K = I = lim I L/K L, C K = C = I/K. To make the shift from local class field theory, we replace the Tate cohomology computations of Ĥ i (Gal(L/K),L ) with Ĥi (Gal(L/K), I K ) = v Ĥ i (Gal(L v /K v ),L v ), where the decomposition into local parts that will allow us to reconstruct from local class field theory is already obvious. Theorem 1.3. (Global class field theory) Let L/K be a finite abelian extension of number fields. We have a global relative Artin map φ L/K : C K /Nm(C L ) Gal(L/K)

3 CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES 3 and there is an order reversing bijection between finite abelian extensions of K and finite index open subgroups of C K. The finite extensions assemble to a global morphism φ K : I K Gal(K ab /K) which in the case of number fields is surjective. We also have the fundamental exact sequence of global class field theory 0 Br(K) v Br(K v ) Q/Z 0. We can now conclude the global Kronecker-Weber Theorem, even though there s a simpler proof which follows directly from the local Kronecker-Weber Theorem. Corollary 1.4. (Kronecker-Weber Theorem) Every abelian extension of Q is contained in a cyclotomic extension. It is now clear that class field theory provides an impressive description of the abelian extensions we initially set off to describe. Still many questions remain open at this point though. Firstly, it is obvious that the global case provides a less explicit construction of the abelian extensions than the local case, and it is this dream of making class field theory explicit that is called Kronecker s Jugendtraum. The other direction one might want to think about is that of non-abelian extensions. Studying the proofs of local and global class field theory it soon becomes apparent that the abelian condition on the Galois groups is essential in the proofs. The study of non-abelian extensions of local and global fields is the Langlands program. We ll now discuss a bit further the case of explicit class field theory. Specifically, Kronecker envisioned in the mid 19th century that one could describe all abelian extensions of Q by adjoining all roots of unity (this turned out to be the Kronecker-Weber Theorem described above), but also that all abelian extension of a quadratic imaginary number field K could be obtained by adjoining the values of a special elliptic function j(z), z K. The former is the Kronecker-Weber theorem, completed by Weber whereas the latter turned out to be almost true and this is exactly the theory of complex multiplication for elliptic curves which we will describe below, providing an analogue of the cyclotomic theory of extensions of Q for quadratic imaginary number fields. 2. CM and Elliptic Curves In this section, fix a quadratic imaginary number field K and let R be an order in K and denote by O K the maximal order, i.e. the ring of integers. We briefly described the motivation behind studying elliptic curves with complex multiplication in the previous chapter and now we will start from the ground up, describing the endomorphism ring of elliptic curves. The moto is that some elliptic curve have more endomorphisms that usual and have very special properties. The uniformisation theory of complex elliptic curves gives us a correspondence between lattices in C up to homothety and isomorphism classes of elliptic curve. Moreover, by the following theorem, we can make an equivalence of categories by describing what happens to the morphisms.

4 4 FRANK GOUNELAS Theorem 2.1. ([Sil92] VI.4.1) Let Λ 1,Λ 2 be lattices in C, we then have a bijection between the set of α C such that αλ 1 Λ 2 and the set of holomorphic maps of complex tori φ : C/Λ 1 C/Λ 2 with φ(0) = 0 by taking a complex number α and sending it to the map φ α (z) = αz mod Λ 2. If E 1,E 2 are elliptic curves corresponding to Λ 1,Λ 2 respectively, we have a bijection between the set of isogenies φ : E 1 E 2 and the sets just described. Proposition 2.2. ([Sil92] VI.5.5) Let E/C be an elliptic curve and let ω 1,ω 2 be generators for the corresponding lattice associated to E. Then either End(E) = Z or Q(ω 1 /ω 2 )/Q is a quadratic extension and End(E) is isomorphic to an order in Q(ω 1 /ω 2 ). Proof. The uniformisation theory of complex elliptic curves tells us that End(E) = R = {α C : αλ Λ} where Λ = Z + τz for τ = ω 1 /ω 2. There exist integers a,b,c,d such that α = a + bτ and ατ = c + dτ. Eliminating τ gives a quadratic equation α 2 +(a+d)α+ab bc = 0 so R/Z is an integral extension. Pick an element α R \Z. We have b 0 and eliminating α gives the quadratic equation bτ 2 (a d)τ c = 0 so Q(τ) is quadratic imaginary. It follows from integrality and R Q(τ) that End(E) is isomorphic to an order of Q(τ). Denote by E (R) the moduli space of elliptic curves with complex multiplication by the order R. We construct elliptic curves with prescribed endomorphism ring the ring of integers O K of an imaginary quadratic number field (or more generally an order in a quadratic imaginary number field). Start with a fractional ideal α in O K, this defines a lattice in C (K is quadratic imaginary) and then the set of a C such that aα is inside α is O K. We know that this lattice defines an elliptic curve and Theorem 2.1 gives us a correspondence between endomorphisms of this elliptic curve and elements a in K such that aα α. Since we are describing lattices up to homothety, cα will give the same elliptic curve up to isomorphism, so we can instead consider an element α Cl(O K ) the class group of K of fractional ideals modulo principal fractional ideals. Theorem 2.3. ([Sil94] II.1.2) Let E Λ = C/Λ be the elliptic curve corresponding to the lattice Λ. There is a well defined free and transitive action of Cl(O K ) on E (O K ) determined by α E Λ = E α 1 Λ. Corollary 2.4. We have an equality # Cl(O K ) = #E (O K ). Some examples of curves with complex multiplication and their corresponding endomorphism ring. (1) Let R = Λ = Z[i]. We have iλ = Λ and hence the coefficient g 3 (Λ) coming from the Eisenstein series (see [Sil92] VI.3) satisfies g 3 (Λ) = g 3 (iλ) = i 6 g 3 (Λ) = g 3 (Λ) so g 3 = 0. Hence the j-invariant is 1728 and we can pick a Q-model of the curve to be y 2 = x 3 + x. (2) Let R = Z[ω]. We similarly get the curve y 2 = x where ω is a cube root of unity. Note both of the above have class number 1 since they are both defined over Q and thus have Q-rational coefficients. We are now in a position to state in what way the theory of complex multiplication provides

5 CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES 5 an analogue of the cyclotomic theory of Q for quadratic imaginary extensions K/Q. First, a technical definition to make the statement of the theorem more accessible. Definition 2.5. For E/C an elliptic curve given by Weierstrass equation y 2 = x 3 +Ax+B and let = 16(4A 3 +27B 2 ) be its discriminant. Define the Weber function φ E : E(C) C as follows AB x(p), if j(e) 0,1728 φ E (P) = A 2 x(p)2, if j(e) = 1728 B x(p)3, if j(e) = 0 Theorem 2.6. ([CSS97] II.20) Let E/C be an elliptic curve with complex multiplication by O K for a quadratic imaginary number field K. The following hold (1) The j-invariant j(e) is an algebraic integer. (2) The field H = K(j(E)) is the Hilbert class field of K. (3) The field K(j(E), {φ E (P) : P E tors }) is isomorphic to K ab the maximal abelian extension of K. Proof. (Sketch) (1) We show first that it is an algebraic number, i.e. j(e) Q. For any field automorphism σ of C and endomorphism φ of E, by acting on coefficients, we get an endomorphism φ σ : E σ E σ and thus an isomorphism End(E) = End(E σ ). From Corollary 2.4, we know that there s finitely many isomorphism classes in E (O K ). Since j(e σ ) = j(e) σ (easy, j is defined in terms of coefficients), we get that j(e) σ takes only finitely many values as σ ranges in Aut(C), hence j(e) Q. Showing that it s in fact an algebraic integer is a bit more involved and we refer the reader to [Sil94] II.6 or V.6.3. (2) Follows from class field theory, see [Sil94] II.4.3. (3) The full statement requires a significant amount of work, which can be found in [Sil92] II.5. One first needs to note that the extension H(E tors )/K is not necessarily an abelian extension so we need to restrict ourselves to the subextension H(φ E (E tors ))/K. We ll describe however how one shows that H(E tors )/H is an abelian extension as it is enlightening as to why we are assuming our curves have complex multiplication. Denote by L m = H(E[m]). Note that H(E tors ) is the composite of the H(E[m]) for m 1 and that we just need to show that L m /H is an abelian extension. Consider the representation ρ : Gal(K/H) Aut(E[m]) given by the condition that ρ(σ)(t) = T σ. For an arbitrary elliptic curve, ρ injects Gal(L m /H) into some subgroup of the automorphism group GL 2 (Z/mZ) of E[m], which is clearly not enough to deduce that it is abelian. Since E has complex multiplication, every endomorphism of E is also defined over H (see [Sil94] II.2.2b). Hence σ Gal(L m /H) commutes with elements in O K when acting on E[m], hence ρ is a homomorphism from Gal(K/H) to the group of O K /mo K -module automorphisms Aut OK /mo K (E[m]) = (O K /mo K ) (see [Sil94] II.1.4b) of E[m], which is abelian and hence the image of ρ is abelian as well.

6 6 FRANK GOUNELAS Despite the vast amount of algebraic data contained in the arithmetic of elliptic curves with complex multiplication, they only form a very small part of the moduli space of elliptic curves over C. We can make this a bit more precise by referencing [Sil94] II.2.1c, which says that the set of isomorphism classes of elliptic curves over C with complex multiplication by O K for K a quadratic imaginary number field as usual is in fact isomorphic to the set of the Q-classes of elliptic curves defined over Q with complex multiplication by O K. In fact, combining what we know about class numbers of quadratic imaginary number fields, we see that there are exactly 13 Q-isomorphism classes of elliptic curves E/Q having complex multiplication (meaning, 13 elliptic curves with CM defined over Q which are not isomorphic over Q, see [Sil92] )! As just described, due to the restrictions imposed on elliptic curves with CM, they represent a class of curves for which much can be said arithmetically. Namely, we have analytic continuation and a function equation for the L-function of such a curve over all number fields, something which remains wide open in the general case and is only known for Q by Wiles proof of the Taniyama-Shimura modularity conjecture. References [CSS97] Gary Cornell, Joseph H. Silverman, and Glenn Stevens, editors. Modular forms and Fermat s last theorem. Springer-Verlag, New York, Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9 18, [Mil08] J.S. Milne. Class field theory (v4.00), Available at [Ser79] Jean-Pierre Serre. Local fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, Translated from the French by Marvin Jay Greenberg. [Sil92] Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, Corrected reprint of the 1986 original. [Sil94] Joseph H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.