Imperial College London Department of Mathematics Rational homotopy theory in arithmetic geometry, applications to rational points Christopher David Lazda June 5, 2014 Supervised by Dr Ambrus Pa´l Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics of Imperial College London and the Diploma of Imperial College London 1 Declaration of Originality I herewith certify that all material in this dissertation which is not my own work has been properly acknowledged. Christopher David Lazda 3 Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. Christopher David Lazda 4 Abstract In this thesis I study various incarnations of rational homotopy theory in the world of arithmetic geometry. In particular, I study unipotent crystalline fundamental groups in therelativesetting,provingthatforasmoothandproperfamilyofgeometricallyconnected varieties f : X → S in positive characteristic, the rigid fundamental groups of the fibres X glue together to give an affine group scheme in the category of overconvergent F- s isocrystals on S. I then use this to define a global period map similar to the one used by Minhyong Kim to study rational points on curves over number fields. Ialsostudyrigidrationalhomotopytypes,andshowhowtoconstructtheseforarbitrary varietiesoveraperfectfieldofpositivecharacteristic. Iprovethattheseagreewithprevious constructionsinthe(log-)smoothandpropercase,andshowthatonecanrecovertheusual rigidfundamentalgroupsfromtheserationalhomotopytypes. Whenthebasefieldisfinite, I show that the natural Frobenius structure on the rigid rational homotopy type is mixed, building on previous results in the log-smooth and proper case using a descent argument. Finally I turn to (cid:96)-adic ´etale rational homotopy types, and show how to lift the Ga- lois action on the geometric (cid:96)-adic rational homotopy type from the homotopy category Ho(dga ) to get a Galois action on the dga representing the rational homotopy type. Q (cid:96) Together with a suitable lifted p-adic Hodge theory comparison theorem, this allows me to define a crystalline obstruction for the existence of integral points. I also study the con- tinuity of the Galois action via a suitably constructed category of cosimplicial Q -algebras (cid:96) on a scheme. 5 Contents Introduction 9 Acknowledgements 12 1 Relative fundamental groups 13 1.1 Relative de Rham fundamental groups . . . . . . . . . . . . . . . . . . . . . 16 1.1.1 The relative fundamental group and its pro-nilpotent Lie algebra . . 18 1.1.2 Towards an algebraic proof of Theorem 1.1.6 . . . . . . . . . . . . . 21 1.2 Path torsors, non-abelian crystals and period maps . . . . . . . . . . . . . . 27 1.2.1 Torsors in Tannakian categories . . . . . . . . . . . . . . . . . . . . . 28 1.2.2 Path torsors under relative fundamental groups . . . . . . . . . . . . 33 1.3 Crystalline fundamental groups of smooth families in char p . . . . . . . . . 35 1.3.1 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.3.2 Frobenius structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.4 Cohomology and period maps . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2 Rigid rational homotopy types 58 2.1 Differential graded algebras and affine stacks . . . . . . . . . . . . . . . . . 59 2.2 Rational homotopy types of varieties . . . . . . . . . . . . . . . . . . . . . . 62 2.2.1 The definition of rigid homotopy types . . . . . . . . . . . . . . . . . 62 2.2.2 Comparison with Navarro-Aznar’s construction of homotopy types . 64 2.2.3 Comparison with Olsson’s homotopy types . . . . . . . . . . . . . . 64 2.2.4 Functoriality and Frobenius structures . . . . . . . . . . . . . . . . . 65 2.2.5 Mixedness for homotopy types . . . . . . . . . . . . . . . . . . . . . 66 2.2.6 Homotopy obstructions . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.3 Overconvergent sheaves and homotopy types . . . . . . . . . . . . . . . . . 75 2.3.1 The overconvergent site . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.3.2 A comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.3 Frobenius structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4 Relative crystalline homotopy types . . . . . . . . . . . . . . . . . . . . . . 82 2.4.1 Another comparison theorem . . . . . . . . . . . . . . . . . . . . . . 85 7 Contents 2.4.2 Crystalline complexes and the Gauss–Manin connection . . . . . . . 86 2.5 Rigid fundamental groups and homotopy obstructions . . . . . . . . . . . . 92 2.5.1 A rather silly example . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3 E´tale rational homotopy types and crystalline homotopy sections 95 3.1 Construction of ´etale rational homotopy types . . . . . . . . . . . . . . . . . 96 3.2 Crystalline sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.4 Relation with other work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.4.1 Pridham’s pro-algebraic homotopy types . . . . . . . . . . . . . . . . 113 3.4.2 The pro-´etale topology of Bhatt/Scholze . . . . . . . . . . . . . . . . 114 Indices 116 Index of definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Index of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Bibliography 118 8 Introduction The topic of this thesis is the study of rational homotopy theory in arithmetic geometry, and my particular interest in the homotopy theory of arithmetic schemes comes from the wish to study rational or integral points. An early indication that homotopy theoretical invariants could give information about rational points was Grothendieck’s anabelian sec- tion conjecture, which says (in particular) that for hyperbolic curves over number fields, rational points are determined by the ´etale fundamental group. In his paper [34], Min- hyong Kim proposed using a unipotent version of the ´etale fundamental group, and a ‘section map’ entirely analogous to the one appearing in Grothendieck’s conjecture, to study integral points on hyperbolic curves over number fields. In the first chapter I look at developing a function field analogue of Kim’s methods, and the main focus is on looking at how the unipotent rigid fundamental group varies in families. If O is some ring of S-integers in a number field, and f : X → Spec(O ) is K,S K,S smooth and proper, then the fact that the Galois action on the (p-adic) unipotent ´etale fundamental group of the generic fibre is unramified away from p and crystalline at places above p can be viewed as saying that this group scheme forms some form of ‘p-adic local system’ on Spec(O ). Moreover, the fibres of this local system at geometric points of K,S Spec(O ) will exactly be the unipotent fundamental group of the fibre of f over that K,S point (of course, some care needs to be taken as to what exactly is meant here at places above p). This suggests the following analogue in positive characteristic. Let C/K be a smooth and proper variety over a global function field K, and spread out to some smooth and proper morphism f : X → S where S is some smooth curve over a finite field. Then the rˆole played in characteristic zero by the (unramified, crystalline) Galois action on the unipotent ´etale fundamental group of the generic fibre should be a certain ‘group scheme of overconvergent F-isocrystals’ on S, whose fibres are the unipotent rigid fundamental groups of the fibres of f. In the first chapter, I show exactly how to construct such an object by proving that a certain sequence of affine group schemes is split exact. This method involves choosing a base point on the curve S, however, a (rather modest) relative form of Tannakian duality gives a rephrasing of this construction in a base-point free way. This also enables me to construct path torsors entirely analogously to those used by Kim to define his period maps, in fact, the definition of the period maps if anything is easier 9 Introduction than in the number field case. This is because the condition that the path torsos over the fundamental group have ‘good reduction’ at places over p is somewhat technical in the number field case, requiring a non-abelian p-adic Hodge theory comparison theorem, whereasinthefunctionfieldsetting,this‘goodreduction’isbuiltintotheveryconstruction of the path torsors. In ordinary homotopy theory, at least for suitably ‘nice’ spaces X, the unipotent fun- damental group π1(X,x)Q can be viewed as the fundamental group associated to a whole ‘homotopy type’ - the ‘rational homotopy type’ of X. In the second chapter, I ask to what extendthesameistruefortheunipotentrigidfundamentalgroup. Tothisend, Ioffersev- eral constructions of commutative differential graded algebras representing ‘rigid rational homotopy types’ - both in the absolute and in the relative case, and prove several com- parison theorems between these construction, and between previous constructions made in special cases by Olsson and Kim/Hain. The main idea is that the rational homotopy type is obtained by simply remembering the multiplicative structure on the cohomology complex,andthecomparisontheoremsarealleffectedbynotingthatthecomparisontheo- rems between the various p-adic cohomology theories respect this multiplicative structure. As a nice application of this construction, I prove that the rigid rational homotopy type of a variety over a finite field admits a weight filtration for the action of Frobenius. In particular this implies that the co-ordinate ring of the unipotent rigid fundamental group also admits a weight filtration - this extends results of Chiarellotto in the smooth case. Ideally, I would like to connect these constructions of rigid rational homotopy types both to the standard construction of the unipotent rigid fundamental group (the absolute case) and to the construction of the first chapter in the relative case. Unfortunately, I am currently only able to effect this comparison in the absolute case, however, I do briefly discussa‘homotopysectionmap’that, withthecorrectcomparisontheorem, shouldrefine the period map defined in the first chapter. If one takes a smooth and proper variety over a global function field K and spread out to some smooth and proper model f : X → S, it is possible to view the target of this homotopy section map as the set of rational homotopy sections with good reduction at all places of S. As in the case of the period map, the fact that these homotopy sections have ‘good reduction’ is built into the very definition of the relative rigid rational homotopy type,sincetheyare,inacertainsense,sectionsofa‘sheafofhomotopytypes’overthebase S. In the third chapter I study the analogous situation for varieties with good reduction over a p-adic field K, that is schemes which arise as the generic fibre of the complement of a relative normal crossings divisor in a smooth and proper scheme X → Spec(O ). K In this case, the fact that the relative homotopy type is in fact some form of ‘sheaf of rational homotopy types’ is expressed by a certain form of Olsson’s non-abelian p-adic 10