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AN INTRODUCTION to P-ADIC HODGE THEORY Contents AN INTRODUCTION TO p-ADIC HODGE THEORY DENIS BENOIS Abstract. These notes provide an introduction to p-adic Hodge theory. They are based on the series of lectures given by the author at the International Center of Theoretical Sciences of Tata Institute in 2019. Contents Introduction 3 1. Local fields. Preliminaries 4 1.1. Non-archimedean fields 4 1.2. Local fields 6 1.3. Ramification filtration 10 1.4. Norms and traces 13 1.5. Witt vectors 14 1.6. Non-abelian cohomology 19 2. Galois groups of local fields 21 2.1. Unramified and tamely ramified extensions 21 2.2. Local class field theory 22 2.3. The absolute Galois group of a local field 22 3. Zp-extensions 24 3.1. The different in Zp-extensions 24 3.2. The normalized trace 27 3.3. Application to continuous cohomology 29 4. Deeply ramified extensions 30 4.1. Deeply ramified extensions 30 4.2. Almost etale´ extensions 34 4.3. Continuous cohomology of GK 36 5. From characteristic 0 to characteristic p and vice versa I: perfectoid fields 38 5.1. Perfectoid fields 38 5.2. Tilting 38 5.3. The ring Ainf(E) 40 5.4. The tilting equivalence 42 6. From characteristic 0 to characteristic p and vice versa II: the field of norms 46 2010 Mathematics Subject Classification. 11F85, 11S15, 11S31, 11G20. Key words and phrases. local field, ramification groups, p-adic representation. 1 2 DENIS BENOIS 6.1. Arithmetically profinite extensions 46 6.2. The field of norms 48 6.3. Functorial properties 52 6.4. Comparison with the tilting equivalence 54 7. `-adic representations 55 7.1. Preliminaries 55 7.2. `-adic representations of local fields (` , p) 58 8. Classification of p-adic representations 59 8.1. The case of characteristic p 59 8.2. The case of characteristic 0 64 9. B-admissible representations 67 9.1. General approach 67 9.2. First examples 69 10. Tate–Sen theory 71 10.1. Hodge–Tate representations 71 10.2. Sen’s theory 73 11. Rings of p-adic periods 76 11.1. The field BdR 76 11.2. The rings Bcris and Bmax 78 11.3. The ring Bst 80 12. Filtered ('; N)-modules 81 12.1. Filtered vector spaces 81 12.2. '-modules 83 12.3. Slope filtration 84 12.4. Filtered ('; N)-modules 86 13. The hierarchy of p-adic representations 87 13.1. de Rham representations 87 13.2. Crystalline and semi-stable representations 88 13.3. The hierarchy of p-adic representations 90 13.4. Comparison theorems 92 14. p-divisible groups 96 14.1. Formal groups 96 14.2. p-divisible groups 101 14.3. Classification of p-divisible groups 103 14.4. p-adic integration on formal groups 105 15. Formal complex multiplication 107 15.1. Lubin–Tate theory 107 15.2. Hodge–Tate decomposition for Lubin–Tate formal groups 109 15.3. Formal complex multiplication for p-divisible groups 110 16. The exponential map 111 16.1. The group of points of a formal group 111 16.2. The universal covering 114 16.3. Application to Galois cohomology 115 16.4. The Bloch–Kato exponential map 117 16.5. Hilbert symbols for formal groups 119 AN INTRODUCTION TO p-ADIC HODGE THEORY 3 17. The weak admissibility: the case of dimension one 122 17.1. Formal groups of dimension one 122 + 'h=p 17.2. Geometric interpretation of (Bcris) 124 References 126 Introduction 0.1. These notes grew out of author’s lectures at the International Center of The- oretical Sciences of Tata Institute in Bangalore in September, 2019. Their aim is to provide a self-contained introduction to p-adic Hodge theory with minimal pre- requisties. The reader should be familiar with valuations, complete fields and basic results in the theory of local fields, including ramification theory as, for example, the first four chapters of Serre’s book [142]. In Sections 3 and 4, we use the lan- guage of continuous cohomology. Sections 15 and 16 require the knowledge of Galois cohomology and local class field theory, as in [142] or [140]. 0.2. Section 1 is utilitarian. For the convenience of the reader, it assembles ba- sic definitions and results from the theory of local fields repeatedly used in the text. In Section 2, we discuss the structure of the absolute Galois group of a lo- cal field. Although only a portion of this material is used in the remainder of the text, we think that it is important in its own right. In Section 3, we illustrate the ramification theory by the example of Zp-extensions. Following Tate, we define the normalized trace map and compute continuous cohomology of Galois groups of such extensions. Krasner [100] was probably the first to remark that local fields of caracteristic p appear as “limits” of totally ramified local fields of characteristic 0 1. In Sections 4- 6, we study three important manifestations of this phenomenon. In Section 4, we introduce Tate’s method of almost etale´ extensions. We consider deeply ramified extensions of local fields and prove that finite extensions of a deeply ramified field are almost etale.´ The main reference here is the paper of Coates and Greenberg [37]. The book of Gabber and Ramero [78] provides a new conceptual approach to this theory in a very general setting, but uses the tools which are beyond the scope of these notes. As an application, we compute continuous Galois cohomology of the absolute Galois group of a local field. In Section 5, we study perfectoid fields following Scholze [130] and Fargues– Fontaine [60]. The connection of this notion with the theory of deeply ramified extensions is given by a theorem of Gabber–Ramero. Again, we limit our study to the arithmetic case and refer the interested reader to [130] for the general treatment. In Section 6, we review the theory of field of norms of Fontaine–Wintenberger and discuss its relation with perfectoid fields. Sections 7-13 are devoted to the general theory of p-adic representations. In Section 7, we introduce basic notions and examples and discuss Grothendieck’s `-adic monodromy theorem. Next we turn to the case ` = p: Section 8 gives an 1See [52] for a modern exposition of Krasner’s results. 4 DENIS BENOIS introduction to Fontaine’s theory of (';Γ)-modules [69]. Here we classify p-adic representations of local fields using the link between the fields of characteristic 0 and p studied in Sections 5-6. In Sections 9-13, we introduce and study special classes of p-adic representations. The general formalism of admissible representa- tions is reviewed in Section 9. In Section 10, we discuss the notion of a Hodge– Tate representation and put it in the frame of Sen’s theory of CK-representations. Here the computation of the continuous Galois cohomology from Section 4 plays a fundamental role. In Section 11-13, we define the rings of p-adic periods BdR, Bcris and Bst and introduce Fontaine’s hierarchy of p-adic representations. Its re- lation with p-adic comparison isomorphisms is quickly discussed at the end of Section 13. In the remainder of the text, we study p-adic representations arising from for- mal groups. In this case, the main constructions of the theory have an explicit description, and p-adic representations can be studied without an extensive use of algebraic geometry. In Section 14, we review the p-adic integration on formal groups following Colmez [38]. A completely satisfactory exposition of this ma- terial should cover the general case of p-divisible groups, which we decided not to include in these notes. For this material, we refer the reader to [64], [66], [39], [30]. In Sections 15-16, we illustrate the p-adic Hodge theory of formal groups by two applications: complex multiplication of abelian varieties and Hilbert pairings on formal groups. In Section 17, we prove the theorem “weakly admissible ) admissible” in the case of dimension one by the method of Laffaille [102]. This implies the surjectivity of the Gross–Hopkins period map. Finally, we apply the + 'h=p theory of formal groups to the study of the spaces (Bcris) ; which play an im- portant role in the theory of Fargues–Fontaine. For further detail and applications of these results, we refer the reader to [60]. 0.3. These notes should not be viewed as a survey paper. Several important as- pects of p-adic Hodge theory are not even mentioned. As a partial substitute, we propose some references for further reading in the body of the text. Aknowledgements. The author is very grateful to Nicola Mazzari for pointing out several inaccuracies in the first version of this text. 1. Local fields.Preliminaries 1.1. Non-archimedean fields. 1.1.1. We recall basic definitions and facts about non-archimedean fields. Definition. A non-archimedean field is a field K equipped a non-archimedean ab- solute value that is, an absolute value j · jK satisfying the ultrametric triangle in- equality: jx + yjK 6 max jxjK;jyjK ; 8x;y 2 K: We will say that K is complete if it is complete for the topology induced by j · jK: AN INTRODUCTION TO p-ADIC HODGE THEORY 5 To any non-archimedean field K; one associates its ring of integers OK = x 2 K j jxjK 6 1 : The ring OK is local, with the maximal ideal mK = x 2 K j jxjK < 1 : The group of units of OK is UK = x 2 K j jxjK = 1 : The residue field of K is defined as kK = OK=mK: Theorem 1.1.2.
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