SUMMER SCHOOL : HIGGS BUNDLES ON P-ADIC CURVES AND
REPRESENTATION THEORYSEPTEMBER 24-28, 2012 AT THE UNIVERSITY OF MAINZ

In 1965, generalising and specifying a former result of Weil, Narasimhan and Seshadri [20]established a bijective correspondence between the set of equivalence classes of irreducible unitary
representations of the fundamental group of a compact Riemann surface X of genus ≥ 2, and theset of isomorphism classes of degree 0 stable vector bundles on X. The correspondence was further
extended to any complex projective smooth variety by Donaldson [9]. The analogue for generallinear representations is due to Simpson, following works by Corlette and Donaldson ; in order toobtain a correspondence of the same type than the one by Narasimhan and Seshadri, we need to
add an extra structure to the vector bundle. It’s the notion of Higgs bundle, introduced first byHitchin for algebraic curves. If X is a scheme, E an OX -module and A an OX -algebra, an A -Higgs
module with coefficients in E is a pair (M , θ) made of an A -module M and an A -linear morphismθ : M → M ⊗OX E such that θ ∧ θ = 0. When X is a complex projective smooth variety, whichis the case considered by Simpson, one exclusively considers OX -Higgs bundles with coefficients inΩ1X/C ; we will simply call them Higgs bundles. But we need later to consider more general Higgs
modules.
Simpson’s main result [24, 25, 26, 27] establishes equivalences of categories between the category of (finite dimensional complex) linear representations of the fundamental group of a complex
smooth projective variety X, the category of vector bundles with integrable connections on X andthe category of semi-stable Higgs bundles on X with vanishing Chern classes (cf. [19]).
Simpson’s results and subsequent developments have led since few years to the quest of a p-adic
analogue. The most advanced approach at the present stage is due to Faltings [12]. It generalises
former results by Tate, Sen and Fontaine, and relies on his theory of almost-étale extensions [11].
It is also a continuation of his works on p-adic Hodge theory, specially those establishing theexistence of Hodge-Tate decompositions [10]. Once achieved, the p-adic Simpson correspondence
should naturally give the best Hodge-Tate type statements in p-adic Hodge theory. At the presentstage, Faltings construction appears to be satisfactory only for curves and, even in this case, many
fundamental questions remain open.
Meanwhile, Deninger and Werner [7, 8, 6] developed a theory of parallel transport for vector bundles over p-adic curves, providing an analogue of the classical theory of Narasimhan andSeshadri.
The summer school at Mainz is an introduction to these p-adic theories, with a particularemphasis on Faltings’ approach. There will be also few lectures on related topics in p-adic Hodge
theory and non-abelian Hodge theory in characteristic p.
Faltings’ approach
Let K be a complete discrete valuation field of characteristic 0, with perfect residue field kof characteristic p > 0, K an algebraic closure of K. We denote by OK the valuation ring ofK, by OK the integral closure of OK in K, by OC the p-adic separated completion of OK and1

2SEPTEMBER 24-28, 2012 AT THE UNIVERSITY OF MAINZby C the field of fractions of OC . Let X be a proper smooth geometrically connected curve overK, x a geometric point of XK . Faltings constructs a fully faithful functor from the categoryof p-adic representations of π1 (XK , x) to the category of (OXC )-Higgs bundles with coefficientsin Ω1X/K (−1). This functor is in fact defined on a strictly larger category than that of p-adic
representations of π1 (XK , x), namely the category of generalised representations of π1 (XK , x), andit induces then an equivalence of categories between this new category and that of (OXC )-Higgs
bundles with coefficients in Ω1X/K (−1). Faltings shows that Higgs modules associated to “true”
p-adic representations of π1 (XK , x) are semi-stables of slope zero. It is expected that all slope zerosemi-stable Higgs modules can be obtained in this way. This statement would correspond to themost difficult part of Simpson’s result in the complex case.
As usually in his work in p-adic Hodge theory, Faltings proceeds by glueing local constructions
on certain affine open subsets of an integral model of X on which he has a certain control andthat he qualifies by small. The local aspect of the theory is in fact developed for schemes of anydimension.
Small representations. The typical example of a small affine scheme considered by Faltings
is an affine scheme Spec(R), étale over the torus Gdm,OK = Spec(OK [T1±1 , . . . , Td±1 ]) and having
integral geometric fibres over Spec(OK ). The fundamental group ∆ of Spec(RK ) (relatively to ageometric generic point) can be explicitly described as follows. Let F be the fraction field of R, F aan algebraic closure of F containing K, F the union of all finite extensions L of F contained in F asuch that the integral closure of R in L is étale over Spec(RK ), R the integral closure of R in Fand R1 = R ⊗OK OK . Then ∆ is the Galois group of F over F ⊗K K. In this setting, generalisedbrepresentations of ∆ are continuous semi-linear representations of ∆ on projective R-modules
ofc
finite type (where for any Zp -module M , we denote M its p-adic separated completion).
Faltings constructs generalised representations from certain Higgs modules. More precisely, let
c1 -Higgs module with coefficients in Ω1c(M, θ) be a R
R/OK (−1) such that the underlying R1 -module is1
free of finite type and that θ is divisible by p2α for a rational number α > p−1; such a Higgs module
is called small. Faltings associates to (M, θ) a p-adically continuous semi-linear representation ϕ ofb The generalised representation thus obtained is small in the sense that it∆ with values in M ⊗ R.
c1Rbbadmits a R-basisof elements that are invariant modulo p2α M ⊗R
c1 R. As a consequence of his purity
theorem for almost étale extensions, Faltings proves that this construction yields an equivalence of
c1 -Higgs
categories between the category of small generalised representations of ∆ and that of small R
module with coefficients in Ω1R/OK (−1). It is the corner stone of the p-adic Simpson correspondence.
The correspondence is moreover compatible with the natural cohomological functors : up to torsion,b
the continuous cohomology of the group ∆ with coefficients in M ⊗R
c1 R is isomorphic to thecohomology of the Koszul complex of (M, θ) (also called the Dolbeault complex of (M, θ)).
Restricting to curves, Faltings develops also the notions of small Higgs modules and small generalised representations with rational coefficients. He obtains again an equivalence of categories.
These results are more generally true for small affine schemes endowed with appropriate log structures.
The most crucial point in this construction, aside from the purity theorem, is the control of its
degree of canonicity. In fact, Faltings gives two constructions, the most useful and better suitedfor globalisation depends on the data of a deformation of the small affine scheme over a certain
infinitesimal thickening A2 (OK ) of OC formerly defined by Fontaine. Inspired by this construction,
but also by the work of Ogus and Vologodsky on an analogue of Simpson’s correspondence inSUMMER SCHOOL : HIGGS BUNDLES ON P-ADIC CURVES AND REPRESENTATION THEORY

3

characteristic p, Abbes and Gros [2] develop another approach to p-adic Simpson correspondence
that generalises Faltings’ approach and that clarifies the relation with Hyodo’s work on Hodge-Tate
local systems [17].
Glueing. The correspondence between small generalised representations and small Higgs modules
described above can be extended for any scheme X over OK endowed with an appropriate logarithmic structure, if we are given a logarithmic deformation of X ⊗OK OC over A2 (OK ). For thispurpose, Faltings uses a topos that he introduced for the proof of the comparison theorems in p-adic
Hodge theory [11]. It turns out that one needs a slightly different version, namely the co-vanishing
topos, which is a variant of oriented products of topos defined by Deligne [3, 18].
Descent. It remains to remove the assumption of smallness. Faltings describes only the case ofcurves. To simplify the exposition, let’s consider a proper semi-stable curve X over OK and fix
a logarithmic deformation of X ⊗OK OC over A2 (OK ). Starting from of a general (OXC )-Higgs
module (E, θ) with coefficients in Ω1X/OK (−1), Faltings shows that, after a finite extension of K,
one can always find a proper morphism f : Y → X such that Y is a semi-stable curve over OK , fK
is a Galois covering and the inverse image f ∗ (E, θ) is small. To this Higgs module is henceforthassociated a small representation M on Y , once fixed a deformation of Y ⊗OK OC over A2 (OK )(which is always possible). It should be then sufficient to descend M . But a priori, there is nonatural descent data on M . To produce such a data, Faltings twists f ∗ (E, θ) by tensor product withan invertible module L . This new module remains small, and the small corresponding generalised
representation is naturally endowed with a descent data. So, it descends to X producing the soughtafter functor. The need of twisting the inverse image of Higgs modules is related to the fact that f
does not necessarily lift to deformations over A2 (OK ). The obstruction for the existence of such alift intervenes in the definition of L . One also needs in the construction to choose an exponential
map on the multiplicative group. Faltings finally shows that one obtains an equivalence of categories
between the category of (OXC )-Higgs modules (E, θ) with coefficients in Ω1X/OK (−1) and that ofgeneralised representations on X with rational coefficients.
Deninger and Werner’s approach
The theory of Deninger and Werner equips pss-vector bundles E of slope zero on smooth projective curves X over p-adic fields with a functorial parallel transport along “étale paths”. Restricted
to closed paths, one obtains representations of the fundamental group on the fibres of the vector
bundle. Here pss stands for “potentially strongly semi-stable reduction”, a somewhat technical notion. The pss-bundles of slope zero on curves form neutral Tannakian categories which are stable
under pullback by finite morphisms. Every pss-bundle is semi-stable and the main open questionis whether the converse holds. If one views a pss-bundle as being equipped with the zero Higgs
field, then the associated representation of the fundamental group on a fibre corresponds to theinverse of Faltings’ construction. However the parallel transport between different fibres provides
a stronger structure. Other topics of interest :– A comparison of the reduction of the p-adic representation attached to (a model) of E withthe representation of the Nori fundamental group (of the special fibre) corresponding tothe reduction of E.
– For pss-bundles of non-zero slope, the construction of a corresponding p-adic representationof a certain central extension of the fundamental group.

4SEPTEMBER 24-28, 2012 AT THE UNIVERSITY OF MAINZ

Reading suggestions for young participants
Beyond the classical theories of étale fundamental groups [16] and local fields [23], the prerequisites for young participants who would like to follow the summer school are on the one hand, thearithmetic theory of p-adic Galois representations as in [15], and on the other hand, the semi-stablereduction theorem for curves [1]. Further readings on p-adic Hodge theory may be useful [10, 11, 14].
For Scholze’s lectures [22], one can consult the video of his lectures at IHÉS on Perfectoid Spaces
and the Weight-Monodromy Conjecture (http ://www.ihes.fr/∼abbes/CAGA/scholze.html).
Références[1] A. Abbes, Réduction semi-stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters...,in “Courbes semi-stables et groupe fondamental en géométrie algébrique”, J.-B. Bost, F. Loeser and M. Raynaud(editors), Progress in Mathematics 187, Birkhäuser (2000), 59-110.
[2] A. Abbes, M. Gros, Sur la correspondence de Simpson p-adique. I : étude locale, prepublication (2011),
arXiv :1102.5466.
[3] A. Abbes, M. Gros, Topos co-évanescents et généralisations, prepublication (2011), arXiv :1107.2380.
[4] F. Andreatta, O. Brinon, BdR -représentations dans le cas relatif, Annales Scientifiques E.N.S. 43 (2010), 279339.
[5] O. Brinon, Représentations p-adiques cristallines et de de Rham dans le cas relatif, Mémoires de la SMF 112(2008).
[6] C. Deninger, Representations attached to vector bundles on curves over finite and p-adic fields, a comparison,
Münster J. of Math. 3 (2010), 29-42.
[7] C. Deninger, A. Werner, Vector bundles on p-adic curves and parallel transport, Ann. Sc. E.N.S. 38 (2005),553-597.
[8] C. Deninger, A. Werner, Vector bundles on p-adic curves and parallel transport II, in Algebraic and arithmeticstructures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math. 58 (2010), Math. Soc. Japan, 1-26.
[9] S. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), 231-247.
[10] G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299.
[11] G. Faltings, Almost étale extensions, dans Cohomologies p-adiques et applications arithmétiques II, Astérisque279 (2002), 185-270.
[12] G. Faltings, A p-adic Simpson correspondence, Adv. Math. 198 (2005), 847-862.
[13] G. Faltings, Crystalline cohomology and p-adic Galois-representations, dans Algebraic analysis, geometry, andnumber theory (Baltimore), Johns Hopkins Univ. Press (1989), 25-80.
[14] J.-M. Fontaine, Le corps des périodes p-adiques, dans Périodes p-adiques, Séminaire de Bures, 1988, Astérisque223 (1994), 59-111.
[15] J.-M. Fontaine, Arithmétique des représentations galoisiennes p-adiques in “Cohomologies p-adiques et applications arithmétiques. III”, Astérisque 295 (2004), 1-115.
[16] A. Grothendieck, Revêtements étales et groupe fondamental, SGA 1, Lecture Notes in Mathematics 224,Springer-Verlag (1971).
[17] O. Hyodo, On variations of Hodge-Tate structures, Math. Ann. 284 (1989), 7-22.
[18] L. Illusie, Produits orientés, dans Travaux de Gabber sur l’uniformisation locale et la cohomologie étale desschémas quasi-excellents, Séminaire à l’École Polytechnique dirigé par L. Illusie, Y. Laszlo et F. Orgogozo(2006-2008).
[19] J. Le Potier, Fibrés de Higgs et systèmes locaux, Séminaire Bourbaki, Exp. 737 (1991), Astérisque 201-203(1992), 221-268.
[20] M.S. Narasimhan, C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. ofMath. 82 (1965), 540-567.
[21] A. Ogus, V. Vologodsky, Non abelian Hodge theory in characteristic p, Pub. Math. IHÉS 106 (2007), 1-138.
[22] , P. Scholze, Perfectoid spaces, preprint (2011), http ://arxiv.org/abs/1111.4914.

SUMMER SCHOOL : HIGGS BUNDLES ON P-ADIC CURVES AND REPRESENTATION THEORY

5[23] J.-P. Serre, Corps locaux (Second edition), Publications de l’Université de Nancago, Hermann, Paris, (1968).
[24] C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867-918.
[25] C. Simpson, Higgs bundles and local systems, Pub. Math. IHÉS, 75 (1992), 5-95.
[26] C. Simpson, Moduli of representations of the fundamental group of a smooth variety I, Pub. Math. IHÉS 79(1994), 47-129.
[27] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety II, Publ. Math.
IHÉS 80 (1994), 5-79.
[28] T. Tsuji, Notes on p-adic Simpson correspondence anf Galois cohomology, preprint (2009).
http ://www.sfb45.de/events/summer-school-higgs-bundles-on-p-adic-curves-and-representation-theory