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The Brauer–Manin Obstruction on Curves R. van Dobben de Bruyn The Brauer–Manin obstruction on curves Mémoire, July 12, 2013 Supervisor: Prof. J.-L. Colliot-Thélène Université Paris-Sud XI Preface This work aims to define the Brauer–Manin obstruction and the finite descent obstructions of [22], at a pace suitable for graduate students. We will give an almost complete proof (following [loc. cit.]) that the abelian descent obstruction (and a fortiori the Brauer–Manin obstruction) is the only one on curves that map non-trivially into an abelian variety of algebraic rank 0 such that the Tate– Shafarevich group contains no nonzero divisible elements. On the way, we will develop all the theory necessary, including Selmer groups, étale cohomology, torsors, and Brauer groups of schemes. 2 Contents 1 Algebraic geometry8 1.1 Étale morphisms........................... 8 1.2 Two results on proper varieties ................... 11 1.3 Adelic points ............................. 17 2 Group schemes 20 2.1 Group schemes............................ 20 2.2 Abelian varieties ........................... 24 2.3 Selmer groups............................. 26 2.4 Adelic points of abelian varieties .................. 32 2.5 Jacobians ............................... 39 3 Torsors 44 3.1 First cohomology groups....................... 44 3.2 Nonabelian cohomology ....................... 45 3.3 Torsors ................................ 47 3.4 Descent data ............................. 53 3.5 Hilbert’s theorem 90......................... 55 4 Brauer groups 58 4.1 Azumaya algebras .......................... 58 4.2 The Skolem–Noether theorem.................... 62 4.3 Brauer groups of Henselian rings .................. 67 4.4 Cohomological Brauer group .................... 69 5 Obstructions for the existence of rational points 74 5.1 Descent obstructions......................... 74 5.2 The Brauer–Manin obstruction................... 77 5.3 Obstructions on abelian varieties.................. 80 5.4 Obstructions on curves........................ 82 A Category theory 86 A.1 Representable functors........................ 86 A.2 Limits................................. 89 A.3 Functors on limits .......................... 91 A.4 Groups in categories......................... 94 B Étale cohomology 100 B.1 Sites and sheaves...........................100 B.2 Čech cohomology...........................103 B.3 Sheafification .............................112 B.4 The étale site.............................116 B.5 Change of site.............................119 B.6 Cohomology..............................125 B.7 Examples of sheaves .........................127 B.8 The étale site of a field........................131 References 134 4 Introduction It is in general a difficult problem to decide whether a variety X over a number field K has any rational points. On the other hand, finding Kv-points for the various completions of K is relatively easy. Through the Hasse principle, for certain classes of varieties the question of whether XpKq is nonempty has become equivalent to the question of whether XpKvq is nonempty for each completion of K. However, there are many classes of varieties known for which there is no Hasse principle, i.e. that have points everywhere locally, but not globally. The proof that they have no global point usually requires some cohomological argument. One of the constructions one can carry out is the formation of the Brauer–Manin set Br X XpAK q : Br X It is a subset of XpAK q containing XpKq, and if one can show that XpAK q “ ?, then in particular X has no rational points. One of the aims of this thesis is to define the Brauer–Manin set and show some of its main properties. At the same time, we will provide certain other obstructions to the existence of rational points. The main theorem (Corollary 5.4.6) is that the Brauer–Manin obstruction is the only obstruction for the existence of rational points on curves C that map non-trivially into an abelian variety A of algebraic rank 0 whose Tate–Shafarevich group contains no non- trivial divisible elements. We develop most of the theory needed to define all the obstructions involved. In particular, we have a lengthy and almost self-contained appendix on étale cohomology. We assume the reader has familiarity with the language of schemes, to a level equivalent to chapters II and III of Hartshorne [10]. Moreover, the reader is assumed to have some knowledge of algebraic number theory, including Galois cohomology. We will at one point use a theorem of global class field theory (Theorem 5.2.5). The language of category theory will be used freely, but we have included an appendix stating some (but possibly not all) of the results we need. This thesis is for a large part based on an article by M. Stoll [22]. We aim at a pace suitable for graduate students in arithmetic geometry, assuming no knowledge of étale cohomology. The treatment of étale cohomology in Appendix B is mostly based on [15]. We tried to minimise the number of external results needed, but sometimes giving the full proof takes us too far afield. 6 Notation Throughout this text, K will denote a field, with separable closure K¯ and ab- solute Galois group ΓK . We will sometimes, by abuse of notation, write K for the scheme Spec K. If K is a number field, then ΩK will denote its set of places. It consists of the f 8 set of finite places ΩK and the set of infinite places ΩK . If S Ď ΩK is a finite subset containing the infinite places, then AK;S will denote the S-adèles, i.e. AK;S “ Kv ˆ Ov: vPS vRS ¹ ¹ The ring of adèles of K is denoted AK : colim A K ; AK “ ÝÑ K;S Ď v S finite vPΩ ¹K 8 where the limit is taken over increasing finite sets S containing ΩK . If S Ď ΩK S is any subset, then AK denotes the adèles with support in S: S AK “ AK X Kv; vPS ¹ K S where the intersection is the one taken in vPΩK v. If is the set of finite or f 8 S the set of infinite places, then we will write±AK and AK respectively for AK . All rings are assumed Noetherian, and all schemes are assumed to be locally Noetherian. We will tacitly assume that morphisms of schemes are locally of finite type, except in the cases where this is obviously false (most notably, a ¯ morphism Spec Kv Ñ X for a completion Kv of K, or the map Spec K Ñ Spec K). A variety X over a field K is a geometrically reduced, separated scheme of finite ¯ ¯ ¯ type over K. The scheme X ˆK K is denoted X; it is a variety over K. Recall that a point x P X is nonsingular (or regular) if OX;x is a regular local ring, and X is smooth at x if pΩX{K qx is free of rank dimpOX;xq. When K is algebraically closed, the two are equivalent, but this is not in general true. Note that x P X is smooth if and only if the corresponding point x¯ P X¯ is nonsingular. A curve over a field K will be a smooth, proper, and geometrically connected (hence geometrically integral) variety over K of dimension 1. A standard result shows that it is in fact projective. If A is an abelian group, then Adiv denotes the subgroup of divisible elements. This need not be a divisible group, as we show in Remark 2.3.16. 7 1 Algebraic geometry We will assume basic familiarity with the language of schemes, for instance following Hartshorne [10]. In this chapter we will prove some additional results that we will need later on. In the final section of this chapter, we introduce some notions that are useful for comparing the K-rational and adelic points for varieties over number fields. 1.1 Étale morphisms Definition 1.1.1. Let f : X Ñ Y be a morphism of schemes that is locally of finite type. Then f is unramified at x P X if, for y “ fpxq, the ideal in Ox generated by my is mx, and the field extension kpyq Ñ kpxq is separable. If f is unramified at all x P X, then f is unramified. Lemma 1.1.2. Let f : X Ñ Y be a morphism of schemes that is locally of finite type. Then f is unramified if and only if ΩX{Y “ 0. Proof. Note that ΩX{Y “ 0 if and only if pΩX{Y qx “ 0 for all x P X. Let x P X be given, and set y “ fpxq. Let V – Spec A be an affine open neighbourhood of y, and let U – Spec B be an affine open neighbourhood of x contained in f ´1V . Firstly, note that pΩX{Y q U – pΩB{Aq~ ˇ pΩ q (by Hartshorne [10], Remark II.8.9.2).ˇ Hence, X{Y x is none other than pΩB{Aqmx . By Matsumura [13], Exercise 25.4, this is the same as ΩBx{Ay . By [loc. cit.], it holds that ΩBx{Ay bAy kpyq “ ΩpBx{my Bxq{kpyq: Moreover, we know that ΩBx{Ay is a finitely generated Bx-module (Hartshorne [10], Corollary II.8.5). Hence, by Nakayama’s lemma, it is zero if and only if ΩpBx{my Bxq{kpyq “ 0. But it is a standard result that a k-algebra of finite type R satisfies ΩR{k “ 0 if and only if R is a finite product of finite separable field extensions of k. Since Bx{myBx is also a local ring, this can only be the case if myBx “ mx and kpxq Ñ kpyq is separable. Hence, f is unramified at x if and only if pΩX{Y qx “ 0. The result follows by considering these conditions for all x P X. Definition 1.1.3. Let f : X Ñ Y be a morphism of schemes. Then f is étale if f is locally of finite type, flat, and unramified. Remark 1.1.4.
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