Local Class Field Theory via Lubin-Tate Theory and Applications Misja F.A. Steinmetz Supervisor: Prof. Fred Diamond June 2016 Abstract In this project we will introduce the study of local class field theory via Lubin-Tate theory following Yoshida [Yos08]. We explain how to construct the Artin map for Lubin-Tate extensions and we will show this map gives an isomorphism onto the Weil group of the maximal Lubin-Tate extension of our local field K: We will, furthermore, state (without proof) the other results needed to complete a proof of local class field theory in the classical sense. At the end of the project, we will look at a result from a recent preprint of Demb´el´e,Diamond and Roberts which gives an 1 explicit description of a filtration on H (GK ; Fp(χ)) for K a finite unramified extension of Qp and × χ : GK ! Fp a character. Using local class field theory, we will prove an analogue of this result for K a totally tamely ramified extension of Qp: 1 Contents 1 Introduction 3 2 Local Class Field Theory5 2.1 Formal Groups..........................................5 2.2 Lubin-Tate series.........................................6 2.3 Lubin-Tate Modules.......................................7 2.4 Lubin-Tate Extensions for OK ..................................8 2.5 Lubin-Tate Groups........................................9 2.6 Generalised Lubin-Tate Extensions............................... 10 2.7 The Artin Map.......................................... 11 2.8 Local Class Field Theory.................................... 13 1 3 Applications: Filtration on H (GK ; Fp(χ)) 14 3.1 Definition of the filtration.................................... 14 3.2 Computation of the jumps in the filtration........................... 15 2 I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define L-series for non-abelian extensions. But for them to agree with the L-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn't, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn't possibly be true. Still, I kept at it, but nothing I tried worked. Not a week went by, for three years, that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I had nothing special to do, so I said, \Well, I try to prove the Reciprocity Law again." So I went out and sat down in the garden. You see, from the very beginning I had the idea to use the cyclotomic fields, but they never worked, and now I suddenly saw that all this time I had been using 1 them in the wrong way and in half an hour I had it. Emil Artin (as recalled by Mattuck in Recountings: Introduction Conversations with MIT Mathematicians, 2009). In this project we will introduce local class field theory via Lubin-Tate theory following Yoshida [Yos08]. Towards the end of the project we will apply the ideas of local class field theory to obtain generalisation of a result from a recent preprint of Demb´el´e,Diamond and Roberts [DDR16]. My aim for this project is not to rewrite Yoshida's paper. Rather I will try to focus more on the intuition behind Yoshida's approach and leave out some details for the sake of brevity. Therefore, instead I will follow the shorter treatments of Neukirch [Neu99] and Serre [Ser67] at times. We should note that Yoshida's approach is heavily based on the approach taken by Iwasawa [Iwa86]. Local class field theory is an old and very important part of algebraic number theory. It was originally studied and proved by Hasse and Schmidt in the 1920s and 30s using global methods, but it was later developed and proved independently using local methods only. The main statements can be summarised in the following theorem. Theorem 1.1. × ab (i) For any local field K; there is a unique homomorphism ArtK : K ! Gal(K =K); characterised by the two properties: (a) If π is a uniformiser of K; then ArtK (π) jKur = FrobK : 0 0× (b) If K =K is a finite abelian extension, then ArtK (NK0=K (K )) jK0 = id: Moreover, ArtK is an isomorphism onto the abelianisation of the Weil group, i.e. × ∼ ab ab Z ArtK : K WK := fσ 2 Gal(K =K) j σ jKur 2 FrobK g (ii) If K0=K is finite separable, then we have a commutative diagram Art 0 K0× K Gal(K0ab=K0) NK0=K K× ArtK Gal(Kab=K) × 0× ∼ 0 ab and ArtK induces an isomorphism K =NK0=K (K ) −−−! Gal((K \ K )=K): Note that the inclusion in the diagram is induced by the natural inclusion of absolute galois groups GK0 ,! GK : 3 Within the scope of this project it was not possible to prove every part of this theorem rigorously. Rather we will focus on the hardest part: we will carefully define the Artin map ArtK and prove that it gives an isomorphism onto the Weil group W (KLT =K) of the maximal Lubin-Tate extension KLT . By invoking base change (i.e. the commutative diagram in part (ii) of the theorem) and the local Kronecker- Weber theorem (i.e. every finite abelian extension of a local field is a Lubin-Tate extension), Theorem 1.1 will follow. Please note that the former of these theorems can be proved directly (see [Yos08, Thm. 5.15]), which gives us a version of local class field theory for Lubin-Tate extensions. 4 2 Local Class Field Theory In this chapter we will carefully construct ArtK of Theorem 1.1 via Lubin-Tate theory. We will first briefly recall some of the theory of formal groups and move on to Lubin-Tate groups. We will define Lubin-Tate extensions and we will explain how to prove local class field theory for these extensions. We will finish by stating how local class field theory for general abelian extensions follows from this. As mentioned before, it is not our intention to give a complete treatment of the material here. We will skip some proofs for the sake of brevity, or just sketch a proof to give the reader some idea of what goes into proving these results. Throughout this chapter, let K be a local field with ring of integers OK ; maximal ideal pK and residue ∗ field k; which has finite order q = qK : For any local field L, let UL = OL denote the units in the ring of integers of L: If E=K is a (possibly infinite) unramified extension, then we will call the completion Eb a complete unramified extension of K: We follow [Neu99, Ch. V], whilst avoiding some of the proofs given there. Later we follow Yoshida [Yos08, x3-5] for the more general theory and full statements. For details, see [Neu99, xV.2 & xV.4] or [Yos08, x3-5]. 2.1 Formal Groups Let O be a ring and O[[X]] the ring of formal power series over O. In practice, O will often be the ring of integers of some local field. Definition 2.1. A (1-dimensional, commutative) formal group over O is a formal power series F (X; Y ) 2 O X; Y with the properties J K (i) F (X; Y ) ≡ X + Y mod deg 2 (i.e. F has no other terms of degree < 2); (ii) F (X; Y ) = F (Y; X) \commutativity"; (iii) F (X; F (Y; Z)) = F (F (X; Y );Z) \associativity". If O = OK , then, given a formal group F , we can define an abelian group structure on pK by setting x +F y := F (x; y) for x; y 2 pK : Note that this makes sense as the power series on the right hand side converges. 5 Definition 2.2. Let F; G be formal groups over O: A homomorphism f : F ! G is a power series 2 f(X) = a1X + a2X + · · · 2 (X) ⊂ O X such that J K f(F (X; Y )) = G(f(X); f(Y )): Note that two homomorphisms compose via composition of power series and f(X) = X acts as the identity. This allows one to naturally define an inverse homomorphism (if it exists) and isomorphisms. The following proposition is now obvious. Proposition 2.3. The homomorphisms f : F ! F of a formal group F over O form a ring End O(F ) in which addition and multiplication are defined by (f +F g)(X) = F (f(X); g(X)); (f ◦ g)(X) = f(g(X)): Definition 2.4. A formal O-module is a formal group F over O together with a ring homomorphism O −−! End O(F ); a 7−! [a]F (X); such that [a]F (X) ≡ aXmod deg 2: From this definition, we it seems natural to define a homomorphism of formal O-modules F; G as a homomorphism of formal groups f : F ! G satisfying f([a]F (X)) = [a]G(f(X)) for all a 2 O: 2.2 Lubin-Tate series Let Kur denote the maximal unramified extension of K and let Ke := Kdur be its completion. Note that p = p O : Let ' 2 Gal(Kur=K) denote the (arithmetic) Frobenius. We can extend ' to K by Ke K Ke e continuity. For any power series F (X ;:::;X ) 2 O X ;:::;X ; let F ' denote the power series which 1 n Ke 1 n arises from F by applying ' to the coefficients.