Arithmetic Fundamental Group Andrew Kobin Spring 2017 Contents Contents Contents 0 Introduction 1 0.1 Topology Review . .1 0.2 Finite Etale´ Algebras . .4 0.3 Locally Constant Sheaves . .5 0.4 Etale´ Morphisms . .8 1 Fundamental Groups of Algebraic Curves 11 1.1 Curves Over Algebraically Closed Fields . 11 1.2 Curves Over Arbitrary Fields . 12 1.3 Proper Normal Curves . 15 1.4 Finite Branched Covers . 16 1.5 The Fundamental Group of Curves . 22 1.6 The Outer Galois Action . 25 1.7 The Inverse Galois Problem . 30 2 Riemann's Existence Theorem 36 2.1 Riemann Surfaces . 36 2.2 The Existence Theorem over 1 ......................... 42 PC 2.3 The General Case . 48 2.4 The Analytic Existence Theorem . 50 3 Scheme Theory 54 3.1 Affine Schemes . 54 3.2 Schemes . 56 3.3 Properties of Schemes . 58 3.4 Sheaves of Modules . 64 3.5 Group Schemes . 70 4 Fundamental Groups of Schemes 73 4.1 Galois Theory for Schemes . 74 4.2 The Etale´ Fundamental Group . 78 4.3 Properties of the Etale´ Fundamental Group . 81 4.4 Structure Theorems . 84 4.5 Specialization and Characteristic p Results . 87 i 0 Introduction 0 Introduction The following notes are taken from a reading course on ´etalefundamental groups led by Dr. Lloyd West at the University of Virginia in Spring 2017. The contents were presented by stu- dents throughout the course and mostly follow Szamuely's Galois Groups and Fundamental Groups. Main topics include: A review of covering space theory in topology (universal covers, monodromy, locally constant sheaves) Covers and ramified covers of normal curves The algebraic fundamental group for curves An introduction to schemes Finite ´etalecovers of schemes and the ´etalefundamental group Grothendieck's main theorems for the fundamental group of a scheme Applications. 0.1 Topology Review There are two key concepts in algebraic topology that, for various reasons, one might want to consider in an algebraic setting. These are covering spaces and fundamental groups, and they are intimately connected. The more familiar concept might be that of the fundamental group, which at the beginning is usually defined in terms of homotopy classes of based loops in a given topological space. To define an algebraic analogue, we will need an alternative perspective on the fundamental group. Let X be a connected, locally simply connected topological space. Definition. A cover of X is a space Y and a map p : Y ! X that is a local homeomorphism. That is, for every x 2 X there is a neighborhood U ⊆ X of x such that p−1(U) is a disjoint union of open sets in Y and p restricts to a homeomorphism on each open set. One consequence of this definition is that for all x; y 2 X, p−1(x) is a discrete space and p p−1(x) ∼= p−1(y). A primary goal in topology is to study and classify all such covers Y −! X. p p0 Definition. A morphism of covers between Y −! X and Y 0 −! X is a map f : Y ! Y 0 making the following diagram commute: f Y1 Y2 p1 p2 X 1 0.1 Topology Review 0 Introduction This defines a category CovX of covers over X. In this category, we will abbreviate HomCovX (Y; Z) by HomX (Y; Z). Example 0.1.1. The unit interval [0; 1] ⊆ R has no nontrivial covers. However, S1 ⊆ C does: for each n 2 Z, the map 1 1 n pn : S −! S ; z 7−! z is a cover. Further, there is a special cover 1 2πit π : R −! S ; t 7−! e such that for every n 2 Z, the following diagram commutes: R S1 π pn S1 All covers of the circle arise in this way. The special property of R ! S1 leads to the notion of a universal cover. Definition. A covering space π : Xe ! X is a universal cover for X if for every other cover p : Y ! X, there is a unique map f : Xe ! Y making the diagram commute: f Xe Y π p X It is equivalent to say that a universal cover is any simply connected cover of X, and one shows easily that universal covers are unique up to equivalence of covers. An important result is that a universal cover exists, under certain mild conditions on X. In topology, the topological fundamental group is defined using homotopy: homotopy classes of loops πtop(X; x) := : 1 in X based at x The universal cover has important connections to this fundamental group. In particular, consider an automorphism α 2 AutX (Xe) = HomX (X;e Xe). Fix x 2 X and a liftx ~ 2 Xe of x. Then πα(~x) = x. Moreover, since Xe is simply connected, any pathx ~ ! αx~ is unique up to top homotopy. This determines a map AutX (Xe) ! π1 (X; x). top Theorem 0.1.2. For any x 2 X, AutX (Xe) ! π1 (X; x) is an isomorphism. 2 0.1 Topology Review 0 Introduction Unfortunately, such a space Xe does not exist in the algebraic world. Thus we describe a slightly different interpretation of the fundamental group. Definition. The fibre functor over x 2 X is the assignment Fibx : CovX −! Sets p (Y −! X) 7−! p−1(x): By the universal property of a universal cover Xe 2 CovX , to give a morphism of covers −1 f : Xe ! Y is the same as to choose a point y = f(~x) 2 p (x). In other words, Fibx is a representable functor, i.e. for x 2 X, there is a natural isomorphism ∼ Fibx(−) = HomX (Xex~; −); where the Hom set consists of morphisms based atx ~. This fibre functor is constructible in algebraic categories, though it fails to be representable. Going further, there is a natural left action of AutX (Xe) on Xe; however, it will be more op convenient to view this as a right action of AutX (Xe) on Xe. This induces a left action of top AutX (Xe) = π1 (X; x) on HomX (X;Ye ): α · f = f ◦ α for any α 2 AutX (Xe); f : Xe ! Y: top This action is called the monodromy action. Often, one views this as an action of π1 (X; x) on the fibre Fibx(Y ) given by lifting paths. In any case, we get a map top π1 (X; x) −! Aut(Fibx); where Aut(Fibx) is the automorphism group of the fibre functor in the following sense. For any functor F : C!D, an automorphism of F is a natural transformation of F that has a two-sided inverse. The set Aut(F ) of all automorphisms of F is then a group under composition. Moreover, Aut(F ) has a natural action on F (C) for any object C 2 C. top Theorem 0.1.3. For all x 2 X, π1 (X; x) ! Aut(Fibx) is an isomorphism. Theorem 0.1.4. Let X be a connected, locally simply connected space and fix x 2 X. Then the fibre functor Fibx defines an equivalence of categories ∼ CovX −!fleft π1(X; x)-setsg with connected covers corresponding to transitive π1(X; x)-sets and Galois covers to coset spaces of Xex by normal subgroups. Proof. For a transitive π1(X; x)-set S, define YS = X=He where H = Stabπ1(X;x)(s) for any point s 2 S. This defines a Galois cover YS ! X and one can extend this to arbitrary π1(X; x)-sets orbitwise for the full correspondence. 3 0.2 Finite Etale´ Algebras 0 Introduction The picture gets more interesting if we restrict ourselves to finite covers. Given such a cover p : Y ! X, there is an exact sequence of groups top −1 1 ! N ! π1 (X; x) ! AutX (p (x)) ! 1 where N is some finite index kernel. This shows that the monodromy action factors through a finite quotient. As a result, this action can be defined on the level of a profinite group, top namely the profinite completion of π1 (X; x): top\ top π1 (X; x) := lim π1 (X; x)=N; − top where the inverse limit is over all finite index subgroups N ≤ π1 (X; x). Corollary 0.1.5. The fibre functor Fibx defines an equivalence of categories ∼ top\ ffinite covers of Xg −!ffinite, continuous π1 (X; x)-setsg: Moreover, the correspondence restricts to ∼ top\ fconnected coversg −!ffinite π1 (X; x)-sets with transitive actiong ∼ top\ and fGalois coversg −!fπ1 (X; x)=N j N an open normal subgroupg: 0.2 Finite Etale´ Algebras Fix a field k and an algebraic closure k¯, which comes equipped with a separable closure ¯ ks ⊆ k. Set Gk = Gal(ks=s) and let L=k be any finite, separable extension. Lemma 0.2.1. Homk(L; ks) is a finite, continuous, transitive Gk-set. Proof. By Galois theory, # Homk(L; ks) = [L : k], so this is finite when the extension is assumed to be finite. The Gk-action is defined by σ·f = σ◦f for σ 2 Gk and f 2 Homk(L; ks); it is routine to verify that this is indeed a group action. Now to show the action is continuous, since Homk(L; ks) is discrete, this is equivalent to showing the stabilizer StabGk (f) is open for each f 2 Homk(L; ks). Notice that StabGk (f) = fσ 2 Gk j σ ◦ f = fg = fσ 2 Gk j σ fixes f(L)g and this is open by Galois theory / the topology of the profinite group Gk. Finally, since L=k is separable, we may pick a minimal polynomial h(t) for a primitive element of L=k.