The Mordell-Weil Theorem

Transcription

1 The Mordell-Weil Theorem Riccardo Brasca 1 Abelian Varieties Let us recall the definition and the basic properties of abelian varieties, all proofs are contained in [6]. Let K denote a field and let K be a fixed algebraic closure of K. If X is a K-scheme and R is a K-algebra, we denote Hom Spec(K) (Spec(R), X) with A(R), and X Spec(K) Spec(R) with X R. The notation X y is used also for the fibre at y Y of a morphism X Y, no problems should arise. The residue field of x X is denoted with k(x). Definition 1.1 An abelian variety over K is a group scheme X that is: separated; of finite type over K; geometrically reduced and connected (i.e. X K is reduced and connected); proper over K. One can think to an abelian variety over K as a complete group variety over K. Some basic facts about varieties imply the following Theorem 1.2 Let A be an abelian variety over K and let K K any field extension. Then A K is an integral and non singular abelian variety over K, in particular any abelian variety is integral and non singular. From now on A will denote an abelian variety over K. By definition the set A(R) is a group for any K-algebra R, in a functorial way. In particular the set A(K) is a group, furthemore if K K is an algebraic extension contained in K, the group structure on A(K ) is the one induced by A(K). By the group law on A, we intend the group law on A(K), for example we 1

2 will speak about the 0 of A, where of course we mean the 0 element of the group A(K) (or of A(K)). Theorem 1.3 Any abelian variety is projective and the group law is commutative. Proof. See for example [2], chapter V. Note that the part about the group law follows from the properness of an abelian variety and it is not difficult, while the projectivity is much harder. A morphism f of K-schemes, between abelian varieties A and B, induces a map, denoted again by f, between each group A(R) and B(R), for any K-algebra R. If f is a morphism as group schemes, each of these maps is a group homomorphism, and it can be proved that this property is equivalent to f(0) = 0. We call such a morphism an homomorphism of abelian varieties. The kernel of an homomorphism f : A B is the fibre of f at the point of B corresponding to 0, so it is A B Spec(K), where the morphism Spec(K) B is the 0 of B(K). The kernel is a closed subgroup of B, of finite type over K. Definition 1.4 An isogeny between abelian varieties A and B defined over K is a surjective homomorphism between A and B such that its kernel is finite over K. It can be proved that if f is an homomorphism then f is an isogeny if and only if its surjective, finite and flat and that isogenies exist only between abelian varieties of the same dimension. The degree of an isogeny f is the degree of its kernel as scheme over K. Note that for an isogeny f : A B the induced homomorphism A(K) B(K) is surjective. Theorem 1.5 Let A be an abelian variety over K. Then for each integer n > 0 there exists an isogeny n: A A such that the induced morphism on A(R), for a K-algebra R, is the multiplication by n. The degree of n is n 2 dim K(A), and it is étale if and only if n does not divide the characteristic of K. Note that if n is étale then the induced map n: A(K) A(K) has finite kernel of order n 2 dim K(A), it can be proved that it is isomorphic, as abstract group, to (Z/nZ) 2 dim K(A). 2

3 2 The Weak Mordell-Weil Theorem Our ultimate goal is to prove that for a number field K and an abelian variety A defined over K, the group A(K) is finitely generated. As in the case of elliptic curves we start by proving the so called weak Mordell-Weil theorem, that states that for an abelian variety A defined over a number field K the group A(K)/n(A(K)) if finite. Let us use the same notations as above and suppose that K is a number field; we denote with R the ring of integers of K: it is well known that R is a Dedekind domain (see [1]). We write K for a generic algebraic extension of K contained in K. The absolute Galois group of K, Gal(K/K ), will be denoted by G K. Fix now an abelian variety A defined over K. For an isogeny ϕ: A A, we write A(K )[ϕ] for the kernel of the induced group homomorphism ϕ: A(K ) A(K ). It is clear that G K acts on each A(K ), and in this way A(K ) becomes a G K -module, that it is easly seen to be discrete (i.e the action is continuous, where G K has the natural topology of a profinite group, and A(K ) is endowed with the discrete topology). For a fixed integer n 1 we have the following exact sequence of G K -modules 0 A(K)[n] A(K) n A(K) 0. (1) Taking G K -cohomology we get 0 A(K)[n] A(K) n A(K) δ H 1 (G K, A(K)[n])... Let us consider now a finite Galois extension K L K, such that L contains the fields of definition of each point of A(K)[n] (this means that each element of A(K)[n] factors throught Spec(L) Spec(K)). The sequence (1) is of G L -modules too, and taking G L cohomology we get 0 A(L)[n] A(L) n A(L) δ H 1 (G L, A(K)[n])... The difference is that now G L acts trivially on A(K)[n] by definition, so H 1 (G L, A(K)[n]) is equal to Hom(G L, A(K)[n]). Note that there is a natural map A(K)/nA(K) A(L)/nA(L), let Φ be its kernel. Using the inflationrestriction exact sequence for G L G K (note that the quotient is finite), we 3

4 have the following commutative diagram Φ A(K)/nA(K) A(L)/nA(L) 0 H 1 (Gal(L/K), A(K)[n]) H 1 (G K, A(K)[n]) H 1 (G L, A(K)[n]) The first row is exact by definition, the sencond is the inflation-restriction exact sequence, the second and third colums are exact by the sequences above and the dotted arrow is defined using the other maps. Being Gal(L/K) and A(K)[n] finite, also H 1 (Gal(L/K), A(K)[n]), and so Φ, is finite. It follows that if we prove the weak Mordell-Weil theorem for L instead of K, we have proved the theorem in general, so from now on we assume that all points of A(K)[n] are defined over K. We have the exact sequence 0 A(K)/nA(K) δ Hom(G K, A(K)[n])... For a point x in A(K)/nA(K), represented by a A(K), we write δ a for δ(x). Suppose that we have found a finite Galois extension K L K such that for every a A(K) and every σ G L G K, δ a (σ) = 0. In this way each δ a would descend to the quotient G K /G L Gal(L/K), and δ would have values in Hom(Gal(L/K), A(K)[n]) that is a finite group. We conclude that in order to prove the weak Mordell-Weil theorem we can find such a finite extension. By general theory δ a (σ) is computed as follows: take any b A(K) such that n(b) = a, then δ a (σ) = σ(b) b for all σ G K. We see that δ a (σ) = 0 if and only if σ(b) = b, so we have to require that σ acts trivially on the field of definition of each b as above. So it is a natural idea to take for L the the compositum of the fields of definition of all b A(K) such that n(b) A(K). We have defined L so that it satisfies the apparantly harder property of our mysterious field (the one about the Galois group), but now we have to prove that it also satisfies the others properties! First of all note that δ induces a map G K Hom(A(K)/nA(K), A(K)[n]), and trivially G L is the kernel of this map, so it is a normal subgroup of G K and K L is Galois. It remains only to show that the extension is finite. What we have just observed suggests to have a look at the injection Gal(L/K) Hom(A(K)/nA(K), A(K)[n]): there is no reason for the second group to be finite, but we see that K L 4

5 is abelian of exponent n (here we use the explicit description, as group, of A(K)[n]). Up to this point we have not used the hypothesis of K being a number field, but we have the following Theorem 2.1 Let K be a number field with ring of integer R, and let S be a finite set of primes of R. For each integer n 1, the maximal abelian extension of K of exponent n unramified outside S is finite. Proof. See [7], Chapter VIII, Proposition 1.6. We have to find a finite set S as in the theorem and show that L is unramified outside S, by the definition L is the compositum of the fields of definition of all b A(K) such that n(b) A(K). Because the compositum of unramified fields is again unramified, we can find a suitable S that works for the field of definition of any fixed b as above. This the hardest part in the proof of the weak Mordell-Weil theorem, and in fact we need some more advanced algebraic geometry. 2.1 End of the proof of the weak Mordell-Weil theorem We know that the morphism A Spec(K) is actually projective, so it is a closed immersion in, say, P n K. As any closed subscheme of the projective space it is determinated by a homogeneous ideal I K[x 0,..., x n ]. Set J = I R[x 0,..., x n ], where R is the ring of integers of the number field K (this is not the ideal of R[x 0,..., x n ] generated by the generators of I with denominators erased). Now let A Spec(R) be the projective scheme corresponding to J. Note that the generic fibre of A is exactly A Spec(K). Being A K A an open immersion, we have that A is isomorphic to an open subscheme of A. Take any non zero prime q of R[x 0,..., x n ]/J, with q R = p: (R[x 0,..., x n ]/J) q is a flat R p -module because it has no torsion (for R p, that is a DVR, these properties are equivalent), so the morphism A Spec(R) is flat, this the reason for the definition of J. What about smoothness? We want to show that there is an open set U Spec(R) such that A Spec(R) U U is smooth. An open subset of Spec(R) is the complement of a finite set of primes, so if we prove that in each of the n + 1 fundamental affine subset of P n R we have to avoid only a finite number of primes we have done. The situation is the following: we have B, a closed subscheme of A n R, with generic ( fibre smooth. By the Jacobian f criterion this means that the matrix i x j )i,j, where the f i s define B, has 5

6 maximal rank when considered as matrix with coefficients in K, it follows that it has maximal rank when viewed as matrix with coefficients in R p /pr p, for all but finitely p Spec(R), as required (look at the determinant of the minor that realizes the rank, it is contained in only finitely many prime ideals of R). So we can assume that we have a smooth and flat morphism π : A U, where U is an open subset of Spec(R), note that A K is again naturally isomorphic to A. Now it s clear what we are trying to do, and it s natural to investigate what happens to the group scheme structure. The fact that A = A K is a group scheme over K is equivalent to the existence of some morphisms which satisfy certain properties, the details are not important because we are only interested on how to extend these morphisms over U. The morphisms are given by some polynomials (of course a finite set) with coefficients in K 1, so they are defined over all but finitely R p and, restricting U if it is necessesarly, we have that A is a group scheme over U. Restricting even more U, we may assume that U = Spec(R ) is affine, where R is a Dedekind ring that is a localization of R, so A is projective R -scheme. The morphism π is proper (it is projective), so π (O A ) is a coherent O U - module, hence π (O A (A))(U) is a finite R -module, it must be free by the flatness of π, and its rank must be 1 because its tensor product with K is Γ(O A, A) = K, it follows that π (O A ) O U. By [3], III Corollary 11.3, π has connected fibres. Now each fibre is a connected group variety, so it is automatically geometrically connected (see [5], pag 8). We have shown that A U is an abelian scheme, where we have the following Definition 2.2 Let S be a scheme. A group scheme A S is an abelian scheme if π a proper and smooth morphism with connected geometric fibres. The fibres of an abelian scheme are by definition abelian varieties, so we can think to A as a family of abelian varieties, with a distinguished one, that is A, and the others will be the reduction of A modulo primes of R. We have constructed A in such a way that its structure of group scheme come from the one of A, so its group law is commutative (this a general fact about abelian schemes), in particular the map n: A A extend to a map n: A A over 1 This is not so clear at all, because there is no equivalence of categories between some kind of K-algebras and projective K-scheme if K is not algebraically closed, but one can reason as follows: our morphisms induce morphisms between the corresponding K- schemes, that are given by polynomials, that, being fixed by G K, have coefficients in K. 6

7 U. We can assume that n is not contained in any of the primes of R, so the map induced on each fibre is étale. Proposition 2.3 With notations as above, the map n: A A is surjective, finite étale of degree n 2 dim K(A). Proof. The morphism is proper because its composition with π, that is separated, is proper. It is quasi finite because this is true on each fibre. It follows that it is a finite morphism, and its degree must be n 2 dim K(A), so if it is étale it will be also surjective. It is flat on each fibre, and by SGA 1, IV, Corollary 5.9, it is flat. Note that until now we have not used that n is not 0 in each of the residue field of U. It must also be unramified because it sheaf of differentials is locally trivial. Now the main result! Theorem 2.4 Let P be any point in A(K), and let Q A(K) such that n(q) = P. Then morphism corresponding to the normalization of R in L, field of definition of Q, is étale. Note that this is what we need (is what we need with R replaced by R, but normalization commutes with localization) in order to complete the proof of the weak Mordell-Weil theorem, indeed we can take as S the complement of the primes in U! Proof. Let π : A U, U = Spec(R ), be as above and let us consider A as an open subscheme of A. For any p prime of R, different from 0, we have the solid arrow commutative diagram Spec(K) Spec(R p) A U Where the morphism Spec(K) A is P. The valutative criterion of properness says that the dotted arrow exists and that it makes the diagram commutative. The image of Spec(R p) in A consists of two points, the generic one and another one, so it is contained in an open affine subset of A, Spec(T ). Furthermore T must be a finitely generated R -algebra, so the morphism T R p extends to a morphism, as R -agebras, T R f for some f R. In other words the morphism Spec(R p) A extends to a morphism, over U, V A, where V is an open subset of U. If V and V are two such open sets, 7

8 with morphisms g and g, then g and g must coincide on the generic point so, being our schemes separated, they must coincide on V V. Glueing all this morphisms we have constructed a section s: U A of π, so we have a commutative diagram P Spec(K) A s U U Let U be the pull back of the diagram A n U s A Then U U is finite étale, because n is. Take now any Q A(K) such that n(q) = P, we have the following solid arrow commutative diagram π Spec(K) Q U A s U A P Spec(K) The dotted arrow exists by universal property. Let U 0 the connected component of U that contains the image of Spec(K). By general theory about étale covering of a normal scheme, we know that U 0 = Spec(R ), where R is the normalization of R is some finite extension of K, say K. We have shown that K contains the field of definition of Q and that R is unramified in K, this concludes the proof (ramification cannot vanish enlarging the extension, look at the ramification index). n 8

9 References [1] John W. S. Cassels and Albrecht Frohlich. Algebraic Number Theory. London Academic Press, [2] Gary Cornell. Arithmetic Geometry. Springer-Verlag, [3] Robin Hartshorne. Algebraic Geometry. Springer-Verlag, [4] Serge Lang. Algebra. Springer, third edition, [5] James Milne. Abelian varieties. Technical report, [6] David Mumford. Abelian Varieties. Bombay Oxford University Press, [7] Joseph H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag,