投射有限群表現之形變理論 - 政大學術集成

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(1)國立政治大學應用數學系碩士學位論文. Deformation Theory of Representations of Profinite Groups 投射有限群表現之形變理論. 碩士班學生: 周惠雯撰指導教授: 余屹正博士中華民國一百零二年一月三日.

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(3) 謝辭滿紙荒唐言, 一把辛酸淚; 都云作者痴, 誰解其中味?. — 曹雪芹. 在政大應數念了兩年半的書, 第一位要感謝的人就是我的指導教授: 余屹正老師。余老師在數學上的熱情是無可比擬的!他在課業上的要求異常嚴格、也很有耐心, 撥出很多時間答覆我的問題, 空出私人的時間幫我上課、教我寫習題, 也告訴我如何尋找參考資料, 時時關心我念書的進度, 逐字逐句批改我的論文, 讓我寫出來的的東西越來越嚴謹。我能順利地拿到學分、通過資格考以及完成論文的撰寫, 完全多虧了余老師, 他使我整個碩士生涯獲益良多、充滿驚奇。第二位要感謝的是陳天進老師。實變函數論的課程內容對我來說相當的困難, 經過陳老師上課的講解, 並且補充了課本沒有的教材, 這些種種對我在消化課文、寫習題上幫助很大, 減輕我在學習過程中的負擔。陳老師教學非常的認真, 上課內容分量很豐富、也很完整, 修過陳老師所開的課之後, 感覺都很充實。最後, 要感謝我的父母給予我的支持, 讓我順利的完成學業。此篇論文獻給所有教過我的老師們、我的家人和朋友。. 周惠雯謹誌于國立政治大學應用數學系中華民國一百零一年十二月. ·v·.

(4) Abstract In this master thesis, we give an exposition of the deformation theory of representations for GL1 and GL2 , respectively, of certain profinite groups. We give rigidity conditions of the fixed representation and verify several conditions for the representability. Finally, we interpret the Zariski tangent spaces of respective universal deformation rings as certain group cohomology and calculate the universal deformation for GL1 .. Keywords: Profinite groups; Representations; Deformations; Universal deformations; Universal deformation rings; Zariski tangent space; Group cohomology. · vi ·.

(5) 摘要在本碩士論文中, 我們闡述了投射有限群表現, 以及其形變理論。我們亦特別研究這些表示在 GL1 和 GL2 之形變, 並且給了可表示化的判定準則。最後, 我們解釋相對應的泛形變環之扎里斯基切空間與群餘調之關連, 並計算了 GL1 的泛形變表現。. 關鍵字:投射有限群; 表現; 形變; 泛形變; 泛形變環; 扎里斯基切空間; 群餘調. · vii ·.

(6) Notations Item. Meaning. ∅. the empty set. X !Y. the set-theoretic difference of X and Y. Z, Q, R, C. integers, rationals, reals, complex numbers. p. a prime integer. G. a profinite group. Zp. the ring of p-adic integers. Qp. the field of p-adic numbers. Gal( L/K ). the Galois group of the field extension L/K. QS. the maximal separable extension of Q unramified outside S. GS. Gal(Q S /Q ). GL n. the general linear group of degree n. G. a connected reductive group (usually stands for GL1 or GL2 in this master thesis). k. a finite field of characteristic p. ρ : G → G( k ). a fixed representation. Hom( A, B). the set of all continuous homomorphisms A → B. W (k) or W. the ring of Witt vectors of k. Ad(ρ ). the two-by-two matrices over k with G-action through ρ and by conjugation. Sets. the category of sets. CNL W. the category of complete noetherian local W-algebras with residue field k. CNL 0W. the subcategory of CNL W consisting of artinian objects. Dρ or D. the deformation functor of ρ. · viii ·.

(7) Contents 謝辭 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vi. 摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. ix. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 2 Profinite Groups and their Representations . . . . . . . . . . . . . . . . . . . .. 6. 2.1 Projective limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2 Profinite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3 Representations of profinite groups . . . . . . . . . . . . . . . . . . . . . .. 9. 2.4 The p-finiteness condition. . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 3 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.1 The ring of Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.2 The deformation functor . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.3 Pro-representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3.4 Schlessinger’s criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.5 The Zariski tangent space and its cohomological interpretation . . . . . . . . .. 25. 4 The Existence of the Universal Deformation . . . . . . . . . . . . . . . . . . . .. 31. 4.1 Verification of condition (H1) . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 4.2 Verification of condition (H2) . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 4.3 Verification of condition (H3) . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 4.4 Verification of condition (H4) . . . . . . . . . . . . . . . . . . . . . . . . .. 34. © Hui-Wen Chou 2012. · ix ·.

(8) 4.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.6 Absolutely irreducible representations . . . . . . . . . . . . . . . . . . . . .. 36. 4.7 Example: the case GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. A Categories and Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. A.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. A.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. A.3 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. B Cohomology for Profinite groups . . . . . . . . . . . . . . . . . . . . . . . . .. 45. B.1 G-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. B.2 Cohomology for profinite groups . . . . . . . . . . . . . . . . . . . . . . .. 46. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. CONTENTS. ·x·.

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(10) Chapter 1 Introduction First Scene: An open place, Thunder and lighting. Enter three witches. Shakespeare, Macbeth, Act I. Nowadays, the idea to study deformations of representations of profinite groups is the achievement of the full Taniyama-Shimura-Weil conjecture proved by Diamond [7], Conrad, Diamond & Taylor [5], and Breuil et al. [2]. However, the concept goes back to the seminal article of Mazur [20]. Mazur’s motivation was to give a conceptual framework for some discoveries of Hida [13] on ordinary families of Galois representations. It was the work of Wiles on Fermat’s Last Theorem which made clear the importance of deformation theory developed by Mazur. The theory was a key technical tool in the proof by Wiles and Taylor-Wiles of Fermat’s Last Theorem; cf. [31, 29]. Mazur’s theory gives one a universal deformation ring which can be thought of as a parameter space for all lifts of a given residual representation (up to conjugation). The ring depends on the residual representation and on supplementary conditions one imposes on the lifts. If the residual representation is modular and if the deformation conditions are such that the p-adic lifts satisfy conditions that hold for modular Galois representations, then one expects in many cases that the natural homomorphism R → T from the universal deformation ring R to a suitably defined Hecke algebra T is an isomorphism. The proof of such isomorphisms, called R = Ttheorems, is at the heart of the proof of Fermat’s Last Theorem. It expresses that all p-adic Galois representation of the type described by R are modular and in particular they arise from geometry. Many refinements of Wiles’ methods have since been achieved and the theory has been vastly © Hui-Wen Chou 2012. ·1·.

(11) generalized to various settings of automorphic forms. R = T theorems, lie at the basis of the proof of the Taniyama-Shimura conjecture by Breuil, Conrad, Diamond and Taylor, the Sato-Tate conjecture by Clozel, Harris, & Taylor [4], Harris, Shepherd-Barron & Taylor [12], and Taylor [28], and the the Serre conjecture by Khare-Wintenberger [14, 15]. The proof of Fermat’s Last Theorem was also the first strong evidence to the conjectures of Fontaine and Mazur [8]. This conjecture says that if a p-adic Galois representation satisfies certain local conditions that hold for Galois representations which arise from geometry, then this representation occurs in the p-adic e´tale cohomology of a variety over a number field. In fact, it is a major motivation for the formulation of the standard conditions on deformation functors. These conditions should (mostly) be local and reflect a geometric condition on a representation. Due to work of Emerton and independently Kisin [16], there has been much progress on the Fontaine-Mazur conjecture over Q. This master thesis focusses solely on the Mazur’s deformation theory, especially for GL1 or GL2 . Generally, one can also consider the representations into certain connected reductive groups; cf. Tiloiune [30]. The obstruction theory and the deformation conditions are left untouched; cf. Mazur [20, 21]. The subject of Fourier transforms is already implicitly such a theory: the exponential function is the equipment one needs to produce a canonical parametrization of the “universal family” of one-dimensional continuous complex unitary representations of the real line R, viewed as a Lie group. For each real number a, putting χ a ( x ) := exp(2πiax ) we have that the universal family of representations of the above type is given parametrically by R → Hom(R, C × ) a #→ χ a . This parameter space itself is again, canonically, the Lie group R, and this miracle has repercussions throughout mathematics. Generally speaking, the “universal parametrization” of all one-dimensional continuous complex unitary representations of any locally compact commutative topological group is treated by. Chap. 1 Introduction. ·2·.

(12) the theory of Pontrjagin. If G is a locally compact topological group, the “Pontrjagin dual” of G ! := Hom(R, C × ) G. is the group parametrizing all degree one continuous unitary C-valued representations of G. ! is again a commutative locally compact The fact that this parameter space of representations, G, topological group is key to the further elaboration of Pontrjagin’s theory.. The more general question of appropriate parametrizations of finite, or infinite, dimensional linear representations of a given type, for a given group, is, of course, one of the great ongoing subjects for our studies. And the natural structure(s) that these parameter spaces come equipped with is, again, key to any further detailed study. Let k be a finite field of characteristic p and let W (k) be the ring of Witt vectors of k. Consider the category CNL W of the complete noetherian local W-algebras A with a surjective local homomorphism ϕ : A → k; the morphisms of CNLW are the local W-algebra homomorphisms commute with the ϕ’s. A representation of the profinite group G is a continuous homomorphism ρ : G → G( A ), where A is a topological, separated, commutative ring and G is a connected reductive group. We say that ( A, ρ") is a lift of ρ over A if A is an object in CNLW and ρ" is a representation for which. the following diagram commutes:. G( A ) "". G. ρ" !!!. ! !! !! ρ. ϕ. !!. ## G(k). where ϕ is the corresponding group homomorphism G( A) → G(k) induced by ϕ : A → k. Two lifts ρ"1 and ρ"2 of ρ over A are said to be strictly equivalent if there exists a matrix M in. ker(G( A) → G(k)) for which ρ"1 ( g) = M" ρ2 ( g) M−1 for every g in G. An equivalence class of. lifts is called a deformation of ρ.. For a representation ρ : G → G(k), the universal deformation ring R is a lift ( R, ρu ) for. Chap. 1 Introduction. ·3·.

(13) which the following universal property holds: for any lift ( A, ρ") of ρ there exists a unique homomorphism ϕ : R → A such that the following diagram commutes: G"" ( R). ρu !!!. G. ! !! !! ρ". ϕ. !!. ## G( A). We say that the profinite group G satisfies the p-finiteness condition Φ p if for all open subgroups G0 ⊂ G of finite index, there are only a finite number of continuous homomorphisms from G0 to Z/pZ. The main theorem is stated as follows: Theorem (Mazur [20], Ramakrishna [23]). Let G be GL1 or GL2 . Suppose that G is a profinite group satisfying the p-finiteness condition Φ p and ρ : G → G(k) is an absolutely irreducible representation. Then there exists a universal deformation ring R in CNLW and a universal deformation ρu of ρ to R, ρ u : G → G( R ) such that any deformation of ρ to a complete noetherian local W-algebra A is obtained from ρu via a unique morphism R → A. However de Smit and Lenstra proved in [6], following an argument due to Faltings, that we can skip the hypothesis of absolute irreducibility if we require the weaker condition Zρ = k for the representation ρ : G → G(k). The structure of this master thesis is organized as follows: Chapter 1 gives a brief review of the theory of profinite groups and their representations. In Chapter 2, we explores the foundations of Mazur’s theory on deformations. We also interpret the Zariski tangent spaces of universal deformation rings as certain group cohomology. Finally, in Chapter 3 we apply the deformation theory to representations and verify the representability conditions of Schlessinger’s criteria. In the process, we shall see where the assumptions for the representability are needed. We also study the 1-dimensional representations, and we will compute the universal deformation ring. The two appendices on categories and functors and on group cohomology provide some fundamental facts we use freely in this master thesis.. Chap. 1 Introduction. ·4·.

(14) Much of the current perspective on deformations of Galois representations is due to work of M. Kisin as is clear to everyone familiar with the topic. Moreover, we found his lecture notes [17] and his paper [18] are very helpful.. ·5·.

(15) Chapter 2 Profinite Groups and their Representations Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.” Michael Atiyah. In this chapter, we give an exposition of the theory of profinite groups and their representations. We also introduce a finiteness condition which is one of the crucial conditions for deformation theory. One can consult Serre’s [27] or Neukirch’s [22] books for further information on profinite groups.. § 2.1. Projective limits A partial order is a binary relation “≤” over a set I which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in I, we have that: (a) a ≤ a (reflexivity); (b) if a ≤ b and b ≤ a, then a = b (antisymmetry); (c) if a ≤ b and b ≤ c, then a ≤ c (transitivity). A set with a partial order is called a partially ordered set. For example, the real numbers ordered by the standard less-than-or-equal relation ≤ and the set of natural numbers equipped with the relation of divisibility are partially ordered sets. © Hui-Wen Chou 2012. ·6·.

(16) A directed set I is a partially ordered set such that for all i, j ∈ I there exists a k ∈ I with i ≤ k and j ≤ k. Definition 2.1.1. Let I be a directed set. (1) A projective system of sets (groups, rings, etc.) indexed by I is a family P = { I, Si , f ij } of sets (groups, rings, etc.) Si and maps (homomorphisms) f ij : S j → Si such that f ii = idSi for each i ∈ I, and f ik = f ij ◦ f jk whenever i ≤ j ≤ k.. (2) We say that S = limi ∈ I Si is a projective limit of the projective system P if it satisfies two ←− conditions: (a) S comes equipped with maps (homomorphisms) f i : S → Si for each i ∈ I such that f i = f ij ◦ f j if i ≤ j. (b) S is universal, i.e., for any other set (groups, rings, etc.) S* and any maps (homomorphisms) g = { gi }i ∈ I : S* → P , there exists a unique homomorphism h : S* → S such that gi = f i ◦ h for all i ∈ I. To show the existence of S, we only have to let S be defined as the following set (group, ring, etc.) S=. #. (σi )i ∈ I. $ % $ $ ∈ ∏ Si $ f ij (σj ) = σi if i ≤ j . $ i∈ I. If the Si are topological spaces and the f ij are continuous maps, then S is a closed subspace of the topological space ∏i ∈ I Si .. § 2.2. Profinite groups A group G is called a topological group if it is a group and a topological space at the same time, and the group operations are continuous. Thus, the product: (a, b) #→ ab and the inverse: a #→ a−1 are continuous maps. The map a #→ a−1 is a homeomorphism since it is an involution. If G is a topological group, for a fixed element a ∈ G, these two maps g #→ ag and g #→ ga are continuous by the continuity of the product. Thus, the left and the right multiplications are §2.2 Profinite groups. ·7·.

(17) homeomorphisms. If the collection U = {U } is a system of open neighborhoods of the identity 1G of G, then aU := { aU | U ∈ U } and U a := {Ua | U ∈ U } are systems of open neighborhoods of a in G. Definition 2.2.1. A topological group G which is the projective limit of finite groups { Gi }i ∈ I , each equipped with the discrete topology, is called a profinite group. As we give these finite groups the discrete topology, they are then compact as topological spaces. By a theorem of Tychonoff, their product with product topology is then compact. Hence, G carries a natural compact Hausdorff topology. Suppose G is a profinite group, so that G = limi ∈ I Gi , and let Ki := ker( f i : G → Gi ). ←− Then Ki is an open subgroup of G and U = {Ki } forms a basis of open neighborhoods of the identity of G. If U ∈ U , then G =. &. a∈ G. aU is an open covering. By the compactness of G,. we can find a finite subcovering such that G =. &h. i =1. aU. This shows that any open subgroup. U ∈ U is of finite index. Since G is compact, it is Hausdorff, and hence. '. U ∈U. U = {1 G }.. The following result gives an intrinsic characterization of profinite groups. Theorem 2.2.2. Let G be a compact Hausdorff group. Then the following assertions are equivalent: (i) G is profinite; (ii) G is totally disconnected, i.e., the connected component of any point is the singleton set consisting only that point; (iii) There is a collection U consisting of open normal subgroups of G that form a full system of neighborhoods of the identity in G. Example 2.2.3. If p is a prime integer, then the rings Z/pn Z, n ∈ N, form a projective system with respect to the canonical projections Z/pn Z → Z/pm Z, n ≥ m. The projective limit Z p := lim Z/pn Z ←− n ∈N. is the ring of p-adic integers. Z p is a pro-p-group, that is, a projective limit of finite p-groups.. §2.2 Profinite groups. ·8·.

(18) Example 2.2.4. The rings Z/nZ, n ∈ N, form a projective system with respect to the projections Z/nZ → Z/mZ for m|n, where the order in N is given by the divisibility m|n. The projective limit ! := lim Z/nZ Z ←− n ∈N. is called the Prüfer ring. By the Chinese remainder theorem and passing to the projective limit, we have a canonical decomposition ! = Z. ∏. Zp.. p: prime. Example 2.2.5. Let L/K be a Galois extension of fields. The Galois group Gal( L/K ) of this extension is, by construction, the projective limit of the Galois groups Gal( Li /K ) of the finite Galois extensions Li /K which are contained in L/K; thus, it is a profinite group. Example 2.2.6. A compact analytic group over the p-adic field Q p is profinite, when viewed as a topological group. In particular, SL n (Z p ), Sp2n (Z p ), GL n (Z p ), . . . are profinite groups.. § 2.3. Representations of profinite groups Let A be a topological, separated, commutative ring and let G be a connected reductive group. Definition 2.3.1. A representation of the profinite group G is a continuous homomorphism ρ : G → G( A ); here we equip G with the profinite topology and G( A) with the linear topology induced by A. For example, let G = GL n ; we will consider the following kinds of topological rings A: (i) Artin representations, i.e., A = C, equipped with its usual topology. Because all compact totally disconnected subgroups of GL n (C ) are finite, these representations have finite image. (Cf. Proposition 2.3.2.) (ii) Mod p representations, i.e., A is a finite field of characteristic p, or more generally, finite rings, like F q (the finite field with q elements) or (Z/p3 Z )[ X, Y ]/(X 4 , (X + Y )2 , Y7 ). §2.3 Representations of profinite groups. ·9·.

(19) We shall always equip them with the discrete topology. These representations arise from elliptic curves and modular forms, and they are the ones that Serre’s conjecture tries to describe. (iii) p-adic representations, i.e., A = Z p , Q p , or more generally a finite dimensional Q p algebra. In this case, A is endowed with its natural topology of normed vector space over Q p , for which it is a topological Q p -algebra. The image of GQ may be infinite (See the Example 2.3.3 below). (iv) Affinoid algebras over Q p . These are the natural coefficients when considering families of representations with coefficients of type (iii), which are exactly the zero-dimensional affinoid algebras. (v) Any other interesting topological ring! Proposition 2.3.2. Suppose A = C and that G is a profinite group. Then every continuous representation ρ : G → G(C ) has finite image. Proof. It suffices to prove in the case G = GL n . We give C n the Euclidean norm: |( x1 , . . . , xn )| = ( ∑i | xi |2 . For each linear transformation T on C n , we define , T , = supv | T (v)| where v runs. through the (n − 1)-dimensional sphere of radius 1. Then , · , is a well-defined norm and the topology of GL n (C ) is given by , · ,.. We will show that the identity matrix 1 is an isolated point; hence, by the compactness the image of ρ is finite. Choosing a sufficiently small open (and hence compact) subgroup H of G, we may assume that ρ( H ) is contained in the open disk of radius 21 centered at 1. Suppose that T = ρ(u) -= 1 for u ∈ H. If all eigenvalues of T equal to 1, then the Jordan canonical form of T has non-diagonal entry. Thus for some large positive integer N, , T N − 1 , > 21 , a contradiction. Then T has an eigenvalue α -= 1. If |α| -= 1, it is obvious that |α N − 1| >. 1 2. for some large integer | N |. If. |α| = 1, the argument of α is small. Thus for an appropriate N, |α N − 1| > 12 , a contradiction. This shows that ρ( H ) = {1 }. Since [ G : H ] < ∞, ρ(G ) is a finite group.. §2.3 Representations of profinite groups. !. · 10 ·.

(20) Example 2.3.3. Let GQ be the absolute Galois group of Q over Q. Take A = Q p and G = G m = GL1 . Let Hom(GQ , G m (Q p )) denote the set of all continuous homomorphisms from GQ into G m (Q p ). We have Hom(GQ , G m (Q p )) = Hom(GQ , Z × p ) = Hom( GQ , µ) × Hom( GQ , 1 + qZ p ), where µ is the torsion subgroup of Z × p and q = 4 if p = 2, q = p if p -= 2. Since 1 + qZ p is a pro-p abelian group, we have p−ab. Hom(GQ , 1 + qZ p ) = Hom(GQ p−ab. where GQ. , 1 + qZ p ). is the largest abelian pro-p quotient of GQ . The Class Field Theory implies that if p−ab. S ⊃ S p := {v place of F| v| p}, then GQ. has positive Z p -rank, hence there exists a continu-. ous representation ρ : GQ → GL1 (Q p ) with infinite image. In this example, the assumption that S contains S p := {v place of F| v| p} is essential in order to get “interesting” S-ramified Galois representation. Remark 2.3.4. Let G be a connected reductive group over a number field F, and let GF,S = Gal( F S /F). Let K be a p-adic field with the ring of integers O . The generalization of class field theory can be formulated as follows. Consider the space YG = Hom(GF,S , G(K )) with the G(K )-action by conjugation. (i) Find a “good” parametrization of the quotient space XG = Hom(GF,S , G(K ))/G(K ). (ii) Find a “large” complete noetherian local O -algebra A and elements ρ in XG ( A) = Hom(GF,S , G( A))/G( A) with “large” image. There are two sources for these. Firstly, the Langlands correspondence. Secondly, the theory of motives via p-adic realization. Langlands’ vision is that these two sources (arithmetic automorphic forms and motives) give the same collection of ρ’s! We have a blueprint for the future development programs as follows. Consider (A) the set of arithmetic automorphic forms on G(A F ) with eigenvalues taking values in certain K0 ; §2.3 Representations of profinite groups. · 11 ·.

(21) (B) YG (K ) = Hom(GF,S , G(K )); (C) Motives over F with coefficients in K0 and good reduction outside of S.. (A) arithmetic automorphic forms """. ?. """ """ """ """ L p """"" """ """ "$$. ## (C) motives # ### # # ## ### # # ## ### M p # # ## %% ## #. (B) YG (K ) = Hom( GF,S , G(K )). From (A) to (B) we have global p-adic Langlands correspondence L p , from (C) to (B) we have the p-adic realization M p of motives. These two operations should be injective. Notice that the sets (A) and (C) are just countable sets, but (B) carries a topology. It is hope that there is a map from (A) to (C) linking these two, hence the image from (A) is contained in the image from (C). For G = GL1 (K ), this is essentially equivalent to class field theory. For GL2 (Q ), this program still remains unproved. For our purposes in this master thesis, we will be mostly interested in p-adic representation G → GL n (K ) where K is a finite extension of Q p , and in families of such representations.. § 2.4. The p-finiteness condition Let G be a profinite group. The pro-p-completion of the profinite group G is G ( p) := lim N G/N ←− where N runs through all closed normal subgroups whose index is a power of p. The p-Frattini quotient of G is the maximal continuous abelian quotient of G which is of exponent p. We recall the pro-p version of the Burnside Basis Theorem: Let G be a pro-p-group, and let Fr(G ) be its pro-p-Frattini quotient. Then any lifting to G of a basis of Fr(G ) as a vector space over Z/pZ is a set of topological generators for G. Definition 2.4.1. We say that the profinite group G satisfies the p-finiteness condition Φ p if for all open subgroups G0 ⊂ G of finite index, there are only a finite number of continuous homomorphisms from G0 to Z/pZ.. §2.4 The p-finiteness condition. · 12 ·.

(22) The following lemma gives several equivalent statements of the p-finiteness condition. Lemma 2.4.2. Let G be a profinite group. The following conditions are equivalent: (i) the pro-p-completion of G is topological finitely generated, (ii) the abelianisation of the pro-p-completion of G is a Z p -module of finite rank, (iii) the p-Frattini quotient of G is finite, (iv) the set of continuous homomorphisms from G to Z/pZ is finite. Proof. Clearly, a set of topological generators of the pro-p-completion becomes a set of generators over Z p in the abelianisation, and becomes a basis of the p-Frattini quotient as a vector space over Z/pZ. Hence, it is clear that (i) implies (ii) and (ii) implies (iii). Since any homomorphism G → Z/pZ must factor through the p-Frattini quotient, (iii) and (iv) are equivalent. The pro-p version of the Burnside Basis Theorem says that if the image in the p-Frattini quotient of a set. { g1 , . . . , gr } of elements of the pro-p-group G( p) is a basis for the quotient as a vector space over Z/pZ, then g1 , . . . , gr topologically generate G ( p) . This shows that (iii) implies (i).. !. Example 2.4.3. For K any number field, let GK = Gal(K/K ) be the absolute Galois group. The structure of the Galois group GK is not so well known. The Kronecker-Weber theorem asserts that ! × induces an isomorphism Gab / Z ! × . Let the natural surjection GQ → Gal(Qcycl /Q ) = Z Q Q S denote as the maximal extension of Q unramified outside a finite set S of primes and let GQ,S be the Galois group of Q S /Q. Here are two famous open problems: Conjecture (Shafarevic).. (i) The absolute Galois group GQab := Gal(Q/Qab ) of Qab is a. free profinite group of countable rank, where Qab is the maximal abelian extension of Q. (ii) Is GQ,S topologically finitely generated? Let us recall the theorem of Hermite and Minkowski, which is the first important fact about GK,S . Theorem 2.4.4 (Hermite-Minkowski). Let K be a finite extension of Q and let S be a finite set of primes. If d is a positive integer, then there are only finitely many extensions L/K of degree d which are unramified outside S. §2.4 The p-finiteness condition. · 13 ·.

(23) An important consequence of the theorem of Hermite and Minkowski is that the set Hom(GQ,S , Z/pZ ) is finite, since each nontrivial homomorphism corresponds to an extension of degree p unramified outside S. Thus, if G0 ⊂ GQ,S is an open subgroup then there exist only finite number of continuous homomorphisms from G0 to Z/pZ. Hence, any finitely ramified Galois groups satisfy the p-finiteness condition Φ p . This is an important example in the deformation theory.. §2.4 The p-finiteness condition. · 14 ·.

(24) Chapter 3 Deformation Theory In these days the angel of topology and the devil of abstract algebra are fighting for every mathematical domain. Hermann Weyl. The basic situation we want to study is as follows. We are given a profinite group G and a representation of G into matrices over a finite field k of characteristic p > 0. We try to understand all possible lifts of this representation to GLn ( A), where A is a complete noetherian local ring with residue k. It is not so clear what “understand all possible lifts” means, and so the main goal of this chapter is to make our question more precise. We begin by recalling some facts of the ring of Witt vectors, then we develop the correct problem of deformation we want to study.. § 3.1. The ring of Witt vectors The materials and the results in this section can be found in the Chapter 2 of Serre’s book [25]. Let K be a field which is complete under a discrete valuation v with residue field k of characteristic p > 0. Let O denote the ring of integers of K and denote the uniformiser of O by '. Then the projection O → k has a unique multiplicative section which associates each λ ∈ k to an element [λ] ∈ O called its Teichmüller representative. In fact, the construction of this section is:. [λ] = lim. n→∞. where λ p. −n. ). p−n λ*. + pn. n. denotes the unique element x ∈ k such that x p = λ and x! denotes an lifting of x in. O . The limit is independent to the choice of the liftings of x and is a well-defined multiplicative section. Denote the set of such multiplicative representatives in O by R.. © Hui-Wen Chou 2012. · 15 ·.

(25) Theorem 3.1.1. With previous notations, every element a ∈ O can be written uniquely as a convergent series. ∞. a=. ∑ [ an ]' n n =0. with [ an ] ∈ R. Moreover, there exists polynomials S0 , S1 , . . . , Sn , . . . and P0 , P1 , . . . , Pn , . . . such that if. ∞. a=. ∞. ∑ [ an ]' n =0. then we have. ∞. a+b =. ∑ n =0. and. ∞. ab =. ∑ n =0. ,. n. and. ∑ [ bn ] ' n ,. b=. n =0. ,. p−n p−n Sn (a0 , . . . , an , b0 , . . . , bn ) ' n. p−n p−n Pn (a0 , . . . , an , b0 , . . . , bn ) ' n .. The last theorem gives more naturally the definition of the Witt vectors which follows: if A is an arbitrary commutative ring with identity, and if a = (a0 , . . . , an , . . .), b = (b0 , . . . , bn , . . .) are elements of AN , we denote W ( A) as the set of such sequences with coefficients in A and equip W ( A) with the laws of composition defined below: a + b = (S0 (a0 , b0 ), . . . , Sn (a0 , . . . , an , b0 , . . . , bn ), . . .) a · b = ( P0 (a0 , b0 ), . . . , Pn (a0 , . . . , an , b0 , . . . , bn ), . . .) These make W ( A) into a commutative ring with identity, called the ring of Witt vectors. Remark 3.1.2. For n ≥ 1, let Wn ( A) = An as a set. If p is invertible in A, we can equip Wn ( A) with a structure of a commutative ring which is isomorphic to An . For the sequence of rings Wn ( A), consider the maps Wn+1 ( A) → Wn ( A). ( a0 , a1 , . . . , a n ) # → ( a0 , a1 , . . . , a n − 1 ).. §3.1 The ring of Witt vectors. · 16 ·.

(26) This is a surjective homomorphism of rings for each n. Then W ( A) / lim Wn ( A); ←− n. thus W ( A) can be viewed as a topological ring. Example 3.1.3. The ring of Witt vectors of the finite field of order p is nothing but the ring of p-adic integers, that is, W (F p ) = Z p . Remark 3.1.4. We have a canonical homomorphism W (k) → O ∞. a = ( a0 , . . . , a n , . . . ) # →. ∑ n =0. ,. p−n. an. -. 'n. This map is always injective and makes O as a W (k)-module of rank e, absolute ramification index. In particular, this map is a bijection if and only if O is unramified.. § 3.2. The deformation functor The strong motivation to study deformations of representations of profinite groups satisfying the p-finiteness condition Φ p is that they play a crucial role in the proof of the modularity conjecture for elliptic curves over Q (work of Wiles, Taylor, Diamond, Breuil, and Conrad [31, 29, 2]). In maximal generality, we begin with a profinite group G and a representation of G into certain matrices over a finite field. The basic question is: can we describe all lifts of this representation to appropriate p-adically complete rings? In order for the theory to work, we need to know that G satisfies the concerned finiteness condition. Let K be a finite extension of Q p with the valuation ring O , the maximal ideal m, and the residue field k of characteristic p. Denote W to be the ring of Witt vectors with coefficients in k. We let G be the algebraic groups GL1 or GL2 .. §3.2 The deformation functor. · 17 ·.

(27) Consider the following topological spaces Y = YG := Hom(G, G(O)) Y := Hom(G, G(k)) X = XG := Y/G(O). X := Y/G(k). where G(O) and G(k) act on Y and Y via conjugation respectively. The reduction homomorphism ϕ : O → k induces serval homomorphisms: ϕ : G(O) → G(k), resp. ϕY : Y → Y, resp. ϕ X : X → X. We have the following commutative diagram: ϕY. Y −−−→   π/. Y   /π. X −−−→ X ϕX. If ρ ∈ Y, then the commutativity of the diagram reads as π (ρ) = π (ρ ). Fix π (ρ) ∈ X. If π (ρ)* is close to π (ρ), there exists ρ* ∈ Y close to ρ such that ρ* = ρ. In other words, the 1 −1 neighborhood U = ϕ− X (π (ρ )) of π (ρ) in X is isomorphic to the quotient of V = ϕY (ρ ) by. its stabilizer Stab(V ) in G(O); Stab(V ) is the inverse image by ϕ of the centralizer of ρ: Stab(V ) = { g ∈ G(O)| gρ* g−1 ∈ V for all ρ* ∈ V }. = { g ∈ G(O)| g¯ ρ¯ g¯ −1 = ρ} = ϕ−1 (Zρ ) ⊂ G(O). So U = V/ϕ−1 (Zρ ) = {ρ* : G → G(O)}/ϕ−1 (Zρ ). Note that ϕ−1 (Zρ ) contains the group ZG (O) of O -points of the center of G. Let CNL W be the category of the complete noetherian local W-algebras A with a surjective local homomorphism ϕ : A → k; the morphisms of CNLW are the local W-algebra homomorphisms commute with the ϕ’s. We will also denote by ϕ the corresponding group homomorphism G( A) → G(k). Definition 3.2.1. Fix a representation ρ : G → G(k). For each object ( A, ϕ) in CNLW , we take H A = ϕ−1 (Zρ ). Consider the covariant functor D = Dρ , called the problem of deformation. §3.2 The deformation functor. · 18 ·.

(28) of ρ: D : CNL W → Sets. ( A, ϕ) #→ U A = VA /H A = VA /ϕ−1 (Zρ ). We call the element in D ( A) a deformation of ρ to A. Remark 3.2.2. For any subgroup H ⊆ G( A), we say that two representations ρi : G → G( A) H. for i = 1, 2 are strictly equivalent with respect to H, written as ρ1 ∼ ρ2 , if ρ2 = hρ1 h−1 for some h ∈ H. Thus, a deformation of ρ to A is in fact a strictly equivalence class of ρ. If ( A, m) is a complete noetherian local ring with residue field k, then we have A = limn A/mn . ←− Definition 3.2.3. We say that a functor F on CNLW is continuous if the canonical morphism F ( A) → limn F ( A/mn ) is an isomorphism. ←− Lemma 3.2.4. The deformation functor D is continuous. Proof. We have to check this map D ( A) → limn D ( A/mn ) is bijective. Note that for each n ←− n + 1 n the map G( A/m ) → G( A/m ) induced by A/mn+1 → A/mn is surjective. For the injectivity, let ρ and ρ* be two representations from G to G( A) such that for each n ≥ 1 there exists an element gn of G( A/mn ) such that for ρn := ρ (mod mn ) and ρ*n := ρ*. (mod mn ), we have gn ρn gn−1 = ρ*n . For each n the set Xn := { gn ∈ G( A/mn )| gn ρn gn−1 = ρ*n } -= ∅, and the transition maps induced by A/mn+1 → A/mn define a projective system of nonempty finite sets Xn+1 → Xn . The projective limit X := limn Xn is therefore nonempty ←− − 1 * and any element g ∈ X satisfies gρg = ρ . For the surjectivity, let {ρn } be a system of representations from G to G( A/mn ) such that for each m ≥ n ≥ 1 there exists gn ∈ G( A/mn ) such that ρm (mod mn ) = gn ρn gn−1 . Starting from ρ1* = ρ1 , one can construct representations ρ*n conjugate to ρn such that ρ*n+1. (mod mn ) = ρ*n by induction. The system {ρ*n : G → G( A/mn )}n defines a representation ρ* with values in G( A) whose class in D ( A) maps to the projective system {π (ρn )} as desired. !. §3.2 The deformation functor. · 19 ·.

(29) Remark 3.2.5. Let CNL 0W be the full subcategory of CNL W whose objects are artinian local rings with residue field k. Notice that the maximal ideal of an artinian local ring is always nilpotent and hence such rings are complete and noetherian. Note also that the objects of CNLW are proobjects of CNL0W , that is, that any object of CNLW is a projective limit of objects of CNL0W . The continuity of deformation functor D shows that D is completely determined by its values on the full subcategory CNL0W . We will use this in a crucial way later, when we use the criteria of Schlessinger for representability, which apply to functors on artinian ring.. § 3.3. Pro-representability The question we want to ask about our deformation functor is whether it is representable. Definition 3.3.1. We say that D is pro-representable by an object R in CNLW if there exists R ∈ Obj(CNL W ) such that the covariant functor h R : CNL0W → Sets A → Homlocalg ( R, A) is naturally isomorphic to D: (a) For any object A ∈ Obj(CNL 0W ), there exist a bijection ι A such that ιA. Homlocalg ( R, A) / D ( A).. (b) For any map α : A → A* , there is a commutative diagram ιA. Homlocalg ( R, A) −−−→ D ( A)     α∗ / /D ( α ). Homlocalg ( R, A* ) −−−→ D ( A* ) ι A*. Remark 3.3.2. Since the deformation functor D is continuous, then it is pro-representable as a functor on CNL 0W if and only if it is representable as a functor on CNLW . §3.3 Pro-representability. · 20 ·.

(30) Lemma 3.3.3. The following two statements are equivalent: (i) D is representable by R; (ii) there exists ξ ∈ D ( R) such that for all η ∈ D ( A) there is a unique morphism α : R → A in CNLW such that D (α)(ξ ) = η. Proof. Suppose that (i) holds. Consider ι R : Homlocalg ( R, R) / D ( R) and ι R (id) = ξ. Since R represents D, for all η ∈ D ( A) there exists a morphism α : R → A in CNLW such that ι A (α) = η and such that the following diagram commutes: ιR. Homlocalg ( R, R) −−−→ D ( R)    D (α) α∗ / /. Homlocalg ( R, A) −−−→ D ( A) ιA. So η = ι A (α) = ι A (α ◦ id) = (ι A ◦ α∗ )(id) = D (α)(ι R (id)) = D (α)(ξ ). This proves (ii). Conversely, suppose (ii) holds. Given ( R, ξ ), we define ι A for each object A of CNLW by: ι A : Homlocalg ( R, A) → D ( A) α #→ D (α)(ξ ) It is a bijection by assumption. Moreover, if A → A* , we have a commutative diagram: ι. A Homlocalg ( R, A) −−− → D ( A)     α∗ / /D ( α ). Homlocalg ( R, A* ) −−−→ D ( A* ) ι A*. §3.3 Pro-representability. · 21 ·.

(31) because D (α) ◦ ι A ( β) = D (α)D ( β)(ξ ) = D (α ◦ β)(ξ ). = D (α∗ ( β))(ξ ) = ι A* (α∗ ( β)) = ι A* ◦ α∗ ( β). This proves (i).. !. Proposition 3.3.4. If ( R, ξ ) exists, it is unique up to a canonical isomorphism. Proof. Let ( R, ξ ) and ( R* , ξ * ) be two pairs, then for any A we have a bijection: ι A : Homlocalg ( R, A) → D ( A) α #→ D (α)(ξ ) ι*A : Homlocalg ( R* , A) → D ( A) α #→ D (α)(ξ * ) Taking A = R* (resp. A = R), we obtain morphisms φ ∈ Homlocalg ( R, R* ) (resp. ψ ∈ Homlocalg ( R* , R)) such that ι R* (φ) = ξ * (resp. ι*R (ψ) = ξ). We now have to show that   φ ◦ ψ = id * R  ψ ◦ φ = id . R. To check second relation, for instance, it suffices to show that ι R (ψ ◦ φ) = ξ. This follows from the calculation ι R (ψ ◦ φ) = D (ψ ◦ φ)(ξ ) = D (ψ) ◦ D (φ)(ξ ). = D (ψ)(ι R* (φ)) = D (ψ)(ξ * ) = ι*R (ψ) = ξ. Similar calculation shows that ι*R* (φ ◦ ψ) = ξ * .. ! §3.3 Pro-representability. · 22 ·.

(32) Definition 3.3.5. The pair ( R, ξ ) is called the universal pair. For any object ( A, ϕ) in CNL W , we set H A = ϕ−1 (Zρ ). For any morphism α : A → A* in CNLW , we define a map, still denoted by α, from U A = Hom(G, G( A))/H A to U A* = Hom(G, G( A* ))/H A* given by π (ρ) #→ π (α ◦ ρ). Corollary 3.3.6. D is representable by R if and only if there exists a continuous homomorphism ρu : G → G( R) such that for any object ( A, ϕ) in CNLW and for any continuous homomorphism ρ : G → G( A) with ϕ(ρ) = ρ there exists a unique local ring homomorphism α : R → A such that the map α : UR → U A sends π (ρu ) to π (ρ). Definition 3.3.7. If D is representable by R, the ring R is called the universal deformation ring of ρ, and the associated representation ρu : G → G( R) is called the universal deformation of ρ.. § 3.4. Schlessinger’s criteria In this section, we first recall a result of Grothendieck for a covariant functor F : CNL 0W → Sets such that F (k) = ξ to be pro-representable and then give useful criteria due to Schlessinger for the pro-representability. Notice that the two categories CNL0W and Sets admit fiber products: Given any two morphisms αi : Ai → A0 in CNL0W , we define their fiber product A3 = A1 × A0 A2 as A3 := {(a1 , a2 ) ∈ A1 × A2 | α1 (a1 ) = α2 (a2 )} m3 := A3 ∩ (m1 × m2 ). We see that ( A3 , m3 ) is an object in CNL0W and the projections β i : A3 → Ai are morphisms in CNL 0W . We put β 0 = α1 ◦ β 1 = α2 ◦ β 2 . Remark 3.4.1. The fiber product of noetherian rings does not need to be noetherian. Indeed, let A = k[[ X, Y ]], B = k and C = k[[ X ]]. Let α : A → C be the map that sends Y to 0 and let β : B → C be the inclusion. The fiber product A ×C B is given by the subring k ⊕ Y · k[[ X, Y ]] in k[[ X, Y ]]. The maximal ideal of A ×C B is Y · k[[ X, Y ]], and the Zariski tangent space of A ×C B may be identified with the k-vector space k[[ X ]], which is infinite dimensional; that is, A ×C B is not noetherian. This is the reason why we consider the smaller category CNL0W . §3.4 Schlessinger’s criteria. · 23 ·.

(33) Given two morphisms αi : Ai → A0 in CNL0W , by the universal property of fiber products we can get a natural map F ( β1 )×F ( β ) F ( β2 ) 0. F ( A1 × A0 A2 ) −−−−−−−−−−−→ F ( A1 ) ×F ( A0 ) F ( A2 ).. (3.1). The result of Grothendieck for a pro-representable covariant functor F is characterized as follows: Theorem 3.4.2 (Grothendieck [9]). The covariant functor F is pro-representable if and only if (i) The map (3.1) is bijective. (ii) F (k[ε]) is a finite set. As Mazur says in [20], the result of Grothendieck is difficult to use because its hypothesis is hard to check for all diagrams A1 $. $$ $$ α1 $$&&. A2 A0. %% %% % ''%% α2. The following criteria of Schlesinger could be viewed as basically a simplification of this result. Theorem 3.4.3 (Schlessinger [24]). Suppose the following four assumptions hold: (H1) If α1 is small (i.e., α1 is surjective and ker(α1 ) is principal and is annihilated by m A1 ), then the map (3.1) is surjective. (H2) If α1 : A1 → A0 is the quotient map k[ε] := k[t]/(t2 ) → k, then the map (3.1) is bijective. (H3) The tangent space tF := F (k[ε]) of F is a finite dimensional k-vector space. (H4) If A1 = A2 , the maps αi : Ai → A0 are the same, and αi is small, then the map (3.1) is bijective. Then F is pro-representable. In particular, there exists an object R in CNL W such that F ( A) = Homlocalg ( R, A) §3.4 Schlessinger’s criteria. · 24 ·.

(34) for every object A in CNL0W . Remark 3.4.4. The finiteness or finite-dimensionality condition is there to guarantee that the representing object is noetherian. (Cf. Corollary 3.5.4.) In the next chapter, we will apply the criteria of Schlessinger to the deformation functor D = Dρ : CNLW → Sets given by D ( A) = {deformations of ρ to A}.. § 3.5. The Zariski tangent space and its cohomological interpretation We have shown that the deformation functor is continuous (cf. Lemma 3.2.4). Suppose that the condition (H2) holds. We can endow the tangent space tD of D, defined by D (k[ε]), with the structure of a k-vector space as follows: Consider the local ring homomorphism which we will simply label “+”: +. k[ε] ×k k[ε] −→ k[ε]. (a + bε, a + b* ε). #→. a + (b + b* )ε.. Let us apply D to it and use the condition (H2). We then obtain a map. + : tD × tD → tD called the addition. It is easy to see that this is a law of an abelian group with zero element given by ρ. Similarly, for λ ∈ k, we see the local ring homomorphism k[ ε] → k[ ε] a + bε #→ a + bλε. §3.5 The Zariski tangent space and its cohomological interpretation. · 25 ·.

(35) Applying D, we then get a map called multiplication by λ. These laws turn tD into a k-vector space. Suppose that A is a complete local W-algebra with residue field k which is given as a projective limit lim. ←− i ∈ I. Ai of a collection of discrete artinian quotients, where i runs through some directed. index set I. We let m and mi be the maximal ideals of A and Ai respectively. Proposition 3.5.1. The following two statements are equivalent: (i) A is noetherian; (ii) dimk (mi /m2i ) is a bounded function of i. Proof. Suppose that A is noetherian. Then m is generated, as an A-ideal, by a finite number d of elements of m. Since m surjects to mi , we have dimk (mi /m2i ) ≤ d for each i, so (i) implies (ii). Now assume that (ii) holds. We first claim that ma = lim. ←− i ∈ I. mia for all a ≥ 0. The assertion. is trivial for a = 0, and we will proceed by induction on a. Assume the statement holds for a and consider the sequence of projective systems 0 → mia+1 → mia → mia /mia+1 → 0. Assumption (ii) implies that mia /mia+1 also has bounded dimension, so the system on the right stabilizes. This implies that its limit is a finite dimensional k-vector space N. By the induction hypothesis we have a short exact sequence 0 → lim mia+1 → ma → N → 0. ←− i. (♥ ). Choose elements b1 , . . . , bl of ma whose images in N form a k-basis of N. For each i we have a surjection Ail → mia , sending ( x1 , . . . , xl ) to x1 b1 + · · · + xl bl . Taking limits we deduce from the induction hypothesis that ma is generated by b1 , . . . , bl as an A-ideal. We now have l ≥ dimk (ma /ma+1 ) ≥ dimk ( N ) = l, so ma+1 is equal to the kernel of the map ma → N. By the sequence (♥) above, this gives the induction step. We now know that A is m-adically complete, and that m is a finitely generated A-ideal. The graded ring G ( A) =. 3. a≥0 m. a /ma+1. is a finitely generated k-algebra, which is noetherian by. §3.5 The Zariski tangent space and its cohomological interpretation. · 26 ·.

(36) Hilbert’s Basis Theorem. By [1], Corollary 10.25, this implies that A is noetherian. This show (i).. !. Definition 3.5.2. Let A be a complete noetherian local W-algebra, and let m A be its maximal ideal. (1) The Zariski cotangent space of A is defined to be t∗A = m A /(m2A , mW ), where (m2A , mW ) = m2A + (image of mW ) A. Note that t∗A is a module over W/mW / k, that is, it is a k-vector space. (2) The Zariski tangent space of A is the dual space of the cotangent space: t A = Homk (t∗A , k).. Remark 3.5.3. Since A is noetherian, t∗A is finite dimensional over k, so that there is no problem with the duality here. Corollary 3.5.4. If the deformation functor D is represented by a complete local W-algebra R, then the ring R is noetherian if and only if t∗R is finite dimensional. Proof. Write R as a projective limit of its discrete artinian quotients Ri . Let mi be the maximal ideal of Ri . Recall that W is noetherian, so that the k-dimension d of mW /m2W is finite. It is clear that dimk (mi /(m2i + mW Ri )) and dimk (mi /m2i ) differ by at most d. Taking limit into account, t∗R is finite dimensional over k if and only if the dimension of mi /m2i is bounded, which by Proposition 3.5.1 is equivalent to R being noetherian.. !. The following lemma gives us a functorial interpretation of the Zariski tangent space: Lemma 3.5.5. If the deformation functor D is represented by an object R in CNLW , then there is a canonical isomorphism of k-vector spaces tR = Homk (t∗R , k) / Homlocalg ( R, k[ε]) = tD . §3.5 The Zariski tangent space and its cohomological interpretation. · 27 ·.

(37) Proof. Let A = k[ε] and let ϕ : R → A. Write ϕ(r ) = ϕ0 (r ) + ϕ1 (r )ε. We have, from ϕ(ab) = ϕ(a) ϕ(b), that ϕ0 (ab) = ϕ0 (a) ϕ0 (b) and ϕ1 (ab) = ϕ0 (a) ϕ1 (b) + ϕ1 (a) ϕ0 (b). Thus, ker( ϕ0 ) = m A = kε. Since ϕ is W-linear, ϕ0 (r ) = r = r (mod mR ), and thus ϕ kills m2R and takes mR W-linearly into kε. For r ∈ W, r = rϕ(1) = ϕ(r ) = r + ϕ1 (r )ε. Hence, ϕ1 kills W. Note that any element of a ∈ R can be written as a = r + x with r ∈ W and x ∈ mR . Thus, ϕ is completely determined by the restriction of ϕ1 to mR , which factors through t∗R . We can then write ϕ : r + x #→ r + ϕ1 ( x )ε and regard ϕ1 as a k-linear map from t∗R into k. Thus ϕ #→ ϕ1 induces a linear map L : Homlocalg ( R, k[ε]) → Homk (t∗R , k). Note that R/(m2R , mW ) = k ⊕ t∗R . For any ψ ∈ Homk (t∗R , k), we extend ψ to R/m2R by declaring its value on k to be zero. Then we define ϕ : R → A by ϕ(r ) = r + ψ(r )ε. Since ε2 = 0, ϕ is a W-algebra homomorphism. In particular, L( ϕ) = ψ; hence, L is surjective. Since any algebra homomorphism killing (m2R , mW ) is determined by its values on t∗R , L is injective.. !. Let ρ : G → GL2 (k) be a representation from G into GL2 (k) and let M2 (k) be the set of all 2 × 2-matrices with entries in k. We let G acts on M2 (k) by the composed map ρ. Ad. G− → GL2 (k) −→ GL(M2 (k)). The k-vector space M2 (k) with the action of G is usually called the adjoint representation of ρ, and is denoted by Ad(ρ ). Proposition 3.5.6. Suppose that the deformation functor D = Dρ is represented by an object R in CNL W . Then there is a canonical isomorphism of k-vector space /. tD −→ H1 (G, Ad(ρ )).. Proof. Let ρ ∈ tD = D (k[ε]) be a deformation of ρ to k[ε]. Since the maximal ideal (ε) of k[ε]. §3.5 The Zariski tangent space and its cohomological interpretation. · 28 ·.

(38) is principal and of square 0, the map /. M2 (k) −→ ker(GL 2 (k[ε]) → GL2 (k)) X. #→. 1 + Xε. is an isomorphism of groups. Thus, we can lift ρ to k[ε] and can compare ρ and ρ. This define an element X (σ) ∈ M2 (k) by ρ(σ) = ρ (σ) + X (σ)ρ (σ)ε. Moreover, σ #→ X (σ) is a 1-cocycle for the adjoint action: ρ(στ ) = ρ(στ ) + X (στ )ρ (στ )ε ρ(σ ) = ρ(σ ) + X (σ )ρ (σ )ε ρ(τ ) = ρ(τ ) + X (τ )ρ (τ )ε ρ(σ)ρ(τ ) = [ρ (σ) + X (σ)ρ (σ)ε] · [ρ (σ)−1 ρ(στ ) + X (τ )ρ (σ)−1 ρ (στ )ε]. = ρ(στ ) + [ X (σ) + Ad ρ(σ)X (τ )]ρ (στ )ε. Conversely, given an 1-cocycle X : G → M2 (k), ρ(σ ) = ρ(σ ) + X (σ )ρ (σ )ε defines a deformation of ρ over k[ε], hence a class ρ ∈ D (k[ε]). Furthermore, if X is a coboundary, we have X (σ) = (Ad ρ(σ) − 1 )Y for some Y ∈ M2 (k). We have the following computation ρ(σ) = ρ(σ) + X (σ)ρ (σ)ε = ρ(σ) + (Ad ρ(σ)Y − Y )ρ (σ)ε. = (1 − Yε) · [ρ (σ) + ρ(σ)(Ad ρ(σ)Y )ε] = (1 − Yε)ρ (σ)(1 + Yε). Hence, We conclude that X is a coboundary if and only if ρ is conjugate to ρ for some element in ker(GL 2 (k[ε]) → GL2 (k)). To complete the proof, we note that the zero element of tD is ρ §3.5 The Zariski tangent space and its cohomological interpretation. · 29 ·.

(39) and it is sent to 0 in H1 (G, Ad(ρ )).. !. Corollary 3.5.7. Suppose that G satisfies the p-finiteness condition Φ p . If the deformation functor D = Dρ is represented by an object R in CNLW , then tD is a finite dimensional k-vector space. Proof. Let G0 = ker(ρ ). This is an open subgroup of G and the action of G0 on Ad(ρ ) is trivial. Note that H0 (G/G0 , H1 (G0 , Ad(ρ ))) = Hom(G0 , M2 (k)) = Hom(G0 , k) ⊗k M2 (k). = Hom(Fr(G0 ), k) ⊗k M2 (k) where Fr(G ) is the pro-p-Frattini quotient of G0 . The inflation-restriction sequence yields the left exact sequence 0 −→ H1 (G/G0 , H0 (G0 , Ad(ρ ))) −→ H1 (G, Ad(ρ )) −→ H0 (G/G0 , H1 (G0 , Ad(ρ ))). The term on the left is finite since G/G0 and H0 (G0 , Ad(ρ )) are finite. The term on the right is finite because of the p-finiteness condition Φ p for G. Hence, this lemma is proved.. §3.5 The Zariski tangent space and its cohomological interpretation. !. · 30 ·.

(40) Chapter 4 The Existence of the Universal Deformation Diese beklagen, daß man heute zu viel abstrakte Mathematik lernen muß, bevor man sinnvoll arbeiten kann. Diese Entwicklung ist zwar zu bedauern, doch darf man nicht u¨ bersehen, daßsie uns andererseits mächtige Hilfsmittel in die Hand gibt, und es erlaubt, komplizierte Sachverhalte einfach und klar darzustellen. Wer diese Methoden ablehnt, wird bei seinen Forschungen meist an der Oberfläche bleiben müssen. Gred Faltings. Let G be GL1 or GL2 . We will give rigidity conditions of the representation ρ : G → G(k) and verify those conditions of Schlessinger’s criteria for our fixed deformation functor D of ρ in this chapter. In the process, we shall see where these assumptions are needed.. § 4.1. Verification of condition (H1) The verification of Schlessinger criteria (H1) will not require any assumption. For any representation ρ : G → G( A) and for any g ∈ G( A), we write g ρ for the representation given by ( g ρ)(σ) = g · ρ(σ) · g−1 . Consider the cartesian square β2. A3 −−−→   β1 /. A2  α / 2. A1 −−−→ A0 α1. © Hui-Wen Chou 2012. · 31 ·.

(41) and the corresponding map D ( A 3 ) → D ( A 1 ) × D ( A 0 ) D ( A 2 ).. (3.1). Assume that α1 is surjective. We must show that (3.1) is surjective. Let (π (ρ1 ), π (ρ2 )) ∈ D ( A1 ) × D ( A2 ) such that D (α1 )(π (ρ1 )) = D (α2 )(π (ρ2 )), where the map π is defined as in §3.2. In other words, we are given two continuous representations ρ i : G → G( Ai ). i = 1, 2,. 1 such that there exists g0 ∈ ϕ− A0 ( Zρ ) such that for all σ ∈ G. ρ20 (σ) =. g0. ρ10 (σ). where ρi0 = αi ◦ ρi : G → G( A0 ) is the push-forward of ρi to A0 . Recall that in algebraic geometry and in commutative algebra, a ring homomorphism f : A → B is said to be formally smooth if it satisfies the following infinitesimal lifting property: Suppose B is given the structure of an A-algebra via the map f . Given a commutative A-algebra C, and a nilpotent ideal J of C, any A-algebra homomorphism B → C/J may be lifted to an A-algebra homomorphism B → C. That is to say, the canonical map Hom A ( B, C) → Hom A ( B, C/J ) ´ ements de is surjective. Formally smooth maps were introduced by Alexander Grothendieck in El´ Géométrie Algébrique [10], IV, Définition (19.3.1). There are also several equivalent definitions of smoothness to be found in EGA IV. The fact that f is smooth if and only if f is locally of finite type and formally smooth is proved in EGA IV, Corollaire (19.5.4). Since G is smooth and α1 is surjective, the map G( A1 ) → G( A0 ) is surjective by the formal 1 smoothness of G. Hence g0 = α1 ( g1 ) for some g1 ∈ G( A1 ) and g1 ∈ ϕ− A1 ( Zρ ), since ϕ A1 =. §4.1 Verification of condition (H1). · 32 ·.

(42) ϕ A0 ◦ α1 . Letting ρ1* =. g1 ρ. 1,. * = ρ . Therefore, ρ* and ρ have the same then we have ρ10 20 2 1. image in G( A0 ), and ρ3 = (ρ1* , ρ2 ) takes values in G( A1 ) ×G( A0 ) G( A2 ) = G( A3 ); moreover, π (ρ3 ) ∈ D ( A3 ) and D ( β i )(π (ρ3 )) = π (ρi ).. § 4.2. Verification of condition (H2) We will study the injectivity properties as stated in conditions (H2) and (H4). If π (ρ3 ) and π (ρ3* ) have the same image, this means we are given two representations ρ3 , ρ3* : G → G( A3 ) H. * for i = 1, 2, with H = ϕ−1 ( Z ). That is, there exist g ∈ H for i = 1, 2, such that ρ3i ∼i ρ3i ρ i i i A i. such that. * ρ3i. =. gi ρ. 3i .. * = Pushing these equalities to A0 , we obtain ρ30. g10 ρ. 30. =. g20 ρ. 30 .. Consider the condition (H2), namely, if A1 → A0 is the quotient map k[ε] → k. In this case, −1 g10 g20 ∈ Zρ .. Note that the centralizer Zρ of ρ is contained in the center ZG of G, where G = G(k); note also that the center ZG of G is GL1 which is formally smooth over W. Since GL1 is smooth and α1 : A1 → A0 is surjective, the map G( A1 ) → G( A0 ) is surjective −1 by the formal smoothness of GL1 . Therefore, we can lift z0 = g10 g20 to z1 ∈ ZG ( A1 ) and g1* = * = g . Hence, we define g = ( g* , g ) ∈ G( A ) × g1 z1 satisfies g10 20 3 1 G ( A 0 ) G( A 2 ) = G( A 3 ). 1 2 1 * We thus have g3 ∈ ϕ− A3 ( Zρ ) and ρ3 =. g3 ρ. 3. as desired, and this shows that (H2) is true.. § 4.3. Verification of condition (H3) ! be the formal group of G defined by Let G0 = ker(ρ ). We let G. ! ( A) := ker ( ϕ : G( A) → G(k)) . G. ! (k[ε]). Suppose that ρ is a lift of ρ to k[ε]. If x ∈ G0 , we have ρ( x ) = 1, and hence ρ( x ) ∈ G §4.2 Verification of condition (H2). · 33 ·.

(43) ! (k[ε]). Two lifts that determine the same map must be Hence, ρ determines a map from G0 to G. identical.. We see that the formal group ! (k[ε]) = {1 + Xε | X ∈ M2 (k)} G. is a p-elementary abelian group and that G0 is an open subgroup of G. By the p-finiteness ! (k[ε]). Hence, D (k[ε]) is a finite condition Φ p , there are only finitely many maps from G0 into G. set. We also have shown that D (k[ε]) itself is a k-vector space in §3.5, and therefore we are done! Remark 4.3.1. This proof relies on the facts that k is a finite field and that the profinite group G satisfies the p-finiteness condition Φ p .. § 4.4. Verification of condition (H4) !G be the formal group of ZG , that is, Z !G = ker(ZG ( A) → ZG (k)). Let Z. Lemma 4.4.1. If Zρ = k, then for any object A in CNL 0W and π (ρ) ∈ D ( A) we have Zρ ∩ ! ( A) ⊆ Z !G ( A). G. Proof. We denote the deformation of ρ to C by ρC for any object C in CNL0W . We let ZρC (C) = { P ∈ Lie(G)(C) | PρC (σ) P−1 = ρC (σ) for all σ ∈ G },. where Lie(G)(C) = M1 (C) or M2 (C) respectively. Since the map A → k is surjective, it factors as a sequence of small extensions. Since Zρ = k, this lemma will follow by induction from the following claim: ! ( B) ⊆ Z !G ( B), then we have Claim: If A → B is small and if ZρB ( B) ∩ G ! ( A) ⊆ Z !G ( A). Zρ A ( A) ∩ G. §4.4 Verification of condition (H4). · 34 ·.

(44) ! ( A). Let x be the image of z in Zρ ( B) ∩ G ! ( B). By our hypothesis, x ∈ Let z ∈ Zρ A ( A) ∩ G B. !G ( B). Let x ∈ Z !G ( A) be a lift of x. Suppose that z #→ x. Then we can write z = x · (1 + tY ) Z. where t is a generator of the kernel A → B and Y ∈ Lie(G)( A).. Since z commutes with the image of ρ A , we must have for every σ ∈ G,. ( x1 + txY )ρ A (σ) = ρ A (σ)( x1 + txY ). This gives Yρ A (σ) = ρ A (σ)Y. Reducing modulo the maximal ideal m A and using the fact Zρ = k, we see that Y is of the form Y = a1 + Y1 where a ∈ A and the entries of Y1 belong to m A . Since A → B is small, we have !G ( A). tm A = 0; it follows that z = x · (ta + 1)1 ∈ Z. !. Suppose that Zρ = k. We will now verify the condition (H4) of Schlessinger’s criteria.. Consider a diagram in CNL0W. β1. A3 −−−→   β1 /. A1  α / 1. A1 −−−→ A0 α1. and assume that α1 : A1 → A0 is surjective. Given π (ρ3 ), π (ρ3 )* ∈ D ( A3 ) with same images in D ( A1 ) ×D ( A0 ) D ( A1 ). In other words, we are given two representations ρ3 = (ρ1 , ρ2 ) : ! ( A1 ) G → G( A3 ) and ρ3* = (ρ1* , ρ2* ) : G → G( A3 ) such that for i = 1, 2 there exists gi ∈ G. we have ρi* =. gi ρ. i.. * . Composing with α , we Let ρ30 = α1 ◦ ρ1 = α1 ◦ ρ2 and similarly for ρ30 1. obtain * ρ30 =. g10. ρ30 =. g20. ρ30. −1 ! ( A0 ). From the previous lemma, Zρ ∩ G ! ( A0 ) consists of the hence z0 = g10 g20 ∈ Zρ30 ∩ G 30. ! ( A0 ) if G = GL1 or GL2 . scalar matrices in G. !G ( A1 ) → Z !G ( A0 ) is surjective by the formal smoothness. Hence there exists We have α1 : Z. ! ( A1 ), we have !G ( A1 ) mapping to z0 such that by putting g* = z1 g1 = g1 z1 ∈ G z1 ∈ Z 1 ! ( A3 ) g3 = ( g1* , g2 ) ∈ G. §4.4 Verification of condition (H4). · 35 ·.

(45) and *. ρ3* = (ρ1* , ρ2* ) = ( g1 ρ1 , g2 ρ2 ) =. g3. ρ3 .. That is, π (ρ3 )* = π (ρ3 ), and the condition (H4) is true.. § 4.5. The main theorem The upshot is: Theorem 4.5.1 (Mazur [20], Ramakrishna [23]). Let G be GL1 or GL2 . Suppose that G is a profinite group satisfying the p-finiteness condition Φ p and ρ : G → G(k) is a continuous representation such that Zρ = k. Then there exists a ring R in CNLW and a deformation ρu of ρ to R, ρ u : G → G( R ) such that any deformation of ρ to a complete noetherian local W-algebra A is obtained from ρu via a unique morphism R → A.. § 4.6. Absolutely irreducible representations The hypothesis that Zρ = k plays an important role in the main theorem. It is of great significance to ask which representations have this property. Definition 4.6.1.. (1) A representation ρ : G → G(k) is said to be reducible if the. representation space has a proper subspace that is invariant under the action of G. (2) It is said to be irreducible if no such subspace exists. (3) We say that ρ is absolutely irreducible if there is no extension k* /k such that ρ ⊗k k* is reducible. Example 4.6.2. The irreducible two-dimensional representation of the symmetric group S3 of order 6 over Q is absolutely irreducible.. §4.5 The main theorem. · 36 ·.

(46) Example 4.6.3. The representation of the circle group by rotations in the plane is irreducible over R, but is not absolutely irreducible. After extending to C, it splits into two irreducible components. This is to be expected, since the circle group is commutative and it is known that all irreducible representations of commutative groups over an algebraically closed field are onedimensional. The following theorem can be found in any standard textbook on group representation theory. For example, see Chapter 1 of Serre’s book [26]. Theorem 4.6.4 (Schur’s Lemma). If the representation ρ : G → G(k) is absolutely irreducible, then Zρ = k. Hence, absolutely irreducible representations have universal deformations. However, there is important other case where ρ is not irreducible but still satisfies Zρ = k. Proposition 4.6.5. Let k be any field, and let V be any representation of G with a G-stable filtration V1 ⊂ V2 ⊂ · · · ⊂ Vn = V such that: (a) Vi +1 /Vi is one-dimensional with G acting by χi ; (b) The χi are distinct; (c) The extension Vi /Vi −1 → Vi +1 /Vi −1 → Vi +1 /Vi is non-split for all i. Then Zρ = k. Proof. Let M ∈ Zρ . We claim that M is a scalar. We first note that V1 is the unique onedimensional subspace on which G acts via χ1 . For if V1* were another, we could build a JordanHolder series V1 ⊂ V1 ∪ V1* ⊂ · · · and thus χ1 would appear at least twice in the Jordan-Holder decomposition which cannot happen since χi are distinct. It follows then that M preserves V1 and by induction preserves the whole flag. Let M act on V1 by multiplication by a. We will show that M = a1. Consider M − a1 : V → V. This element M − a1 is also in Zρ . Since M − a1 |V1 = 0, it factors as a morphism T : V/V1 → V. §4.6 Absolutely irreducible representations. · 37 ·.

(47) By induction, the induced map V/V1 → V/V1 is G-invariant which is multiplication by a scalar b. If b -= 0, then T |V2 would give a splitting of the extension where i = 1 and so we can assume b = 0. Thus, T is actually a G-invariant map V/V1 → V. If T = 0, then we are done, else let Vi be the first subspace on which it is non-trivial. Then T : Vi /Vi −1 → V1 is a G-module. !. isomorphism, contradiction.. § 4.7. Example: the case GL1 For G = GL1 and ρ : G → k× , we see that the assumptions Zρ = k and the center of GL1 is formally smooth over W are trivially fulfilled. We will compute R = Ru and ρu in this section. Consider the deformation ρ of ρ to A, i.e., a character ρ : G → A× . Since A ∈ Obj(CNL W ), the reduction morphism A → k has a multiplicative lifting ω A : α. k× → A× called the Teichmüller lifting which is functorial: if A → B → k, then α ◦ ω A = ωB . Write ρ(σ) = ω ◦ ρ (σ) · 8ρ9(σ) with 8ρ9(σ) ≡ 1 (mod m). Since 1 + m is pro-p abelian, the character 8ρ9 factors through the maximal p-abelian quotient G p,ab of G. We define ψρ : W [[ G p,ab ]] → A as the unique local W-algebra homomorphism such that for all σ ∈ G p,ab ψρ ([σ]) = 8ρ9(σ) where [σ] denotes the corresponding element of σ in the group ring W [[ G p,ab ]]. Proposition 4.7.1. For G = GL1 , the universal pair ( Ru , ρu ) is given by Ru = W [[ G p,ab ]]. §4.7 Example: the case GL1. · 38 ·.

(48) and ρu : G → W [[ G p,ab ]] σ #→ ω (ρ (σ)) · [σ p,ab ] where σ #→ σ p,ab is the projection G → G p,ab . Proof. By the p-finiteness condition Φ p , we know that G p,ab is finitely generated as a Z p module. If r is the number of generators, then W [[ G p,ab ]] is a quotient of the power series ring W [[t1 , . . . , tr ]] and, therefore, is a complete noetherian local W-algebra. Take any deformation ρ of ρ to A, i.e., ρ : G → A× , we get a local W-algebra homomorphism ψρ : W [[ G p,ab ]] → A uniquely determined by the condition ψρ ([σ]) = 8ρ9(σ). But we have ψρ (ρu (σ)) = ψρ (ω (ρ (σ)) · [σ p,ab ]). = ω (ρ (σ))ψρ ([σ p,ab ]) = ω (ρ (σ))8ρ9(σ) = ρ ( σ ), that is, ψρ ◦ ρu = ρ. Thus W [[ G p,ab ]] is the universal deformation ring and ρu defined above is the universal deformation of ρ.. !. Remark 4.7.2. If we fix an topological group isomorphism G p,ab / H × Zrp where H is a finite group, we obtain a local W-algebra isomorphism W [[ G p,ab ]] / W [[t1 , . . . , tr ]][ H ]. That is, the universal deformation ring is the group algebra of a finite group over an Iwasawa algebra, ring of formal power series in r variables over W. §4.7 Example: the case GL1. · 39 ·.

(49) Appendix A Categories and Functors It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out. Emil Artin, Geometric Algebra. The language of categories and functors is a particularly convenient way to think about the deformation theory. We will introduce the concept of categories to serve as a useful tool and to provide a general context for dealing with a number of different mathematical situations in this master thesis. The more details and materials are contained in the book of Mac Lane [19].. § A.1. Categories Definition A.1.1. A category C is defined by the following two data: • a collection of objects of C, denoted by Obj(C); • For any A and B in Obj(C), there is a set HomC ( A, B) and referred to as the set of morphisms from A into B. satisfying the following rules: (a) For any A, B and C in Obj(C), there is a rule of composition for morphisms, i.e., a mapping: HomC ( A, B) × HomC ( B, C) → HomC ( A, C) : ( f , g) #→ g ◦ f ; f. g. h. (b) (Associativity). For any three morphisms: A → B → C → D, we have h ◦ ( g ◦ f ) =. (h ◦ g) ◦ f ;. © Hui-Wen Chou 2012. · 40 ·.