Vector bundles on padic curves and parallel transport. This result leads to an extension of the parallel transport theory by deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. Jeanmarc fontaine joint work in progress with laurent fargues created date. We construct a functor from the category of p adic etale local systems on a smooth rigid analytic variety x over a p adic eld to the category of vector bundles with an integrable connection on its \base change to b dr, which can be regarded as a rst step towards the soughtafter p adic riemannhilbert correspondence. For a variety in characteristic p with a lift to characteristic 0, fontaine conjectured and. Our method relies on the classification of vector bundles over the fargues fontaine curve. The proof combines a dynamical argument on banachcolmez spaces with an optimal existence theorem.
So while the bundles edo not extend to smin, they do on scholzes limit space pmin. We shall extend some of these results to g vector bundles. In lectures 1215, we study shtukas with one leg, and their connection to p divisible groups and p adic hodge theory. Contents 1 introduction 2 2 formal vector bundles with marked sections.
An important technique in such analyses is the general rankdegree formalism of harder and narasimhan 25, which applies to vector bundles on any curve. Geometry of mmf p adic geometryresults some geometric notation set 0n or 1n, k c. A topological ring r is called a tate ring if it is fadic and contains a topologically. The deviation of an irregular singular connection from the regular singular case, or equivalently the complexity of the stokes data, is measured by the slope of the connection. For example, if we want to study galksepkrepresentations, we can look at the galois action on the farguesfontaine curve. Peter scholzes lectures on padic geometry, fall 2014. Z p, which is a z p module that comes equipped with an action of galkk, for a. This paper is a report on joint work with christopher deninger published in dewe1, dewe2 and dewe3. The farguesfontaine curve and diamonds dapr es fargues. In general, if we have a smooth proper family of algebraic varieties, the etale cohomology of the bers yields both adic and p adic etale local systems on the base. In section 2 we define a category bacp of vector bundles on acp which contains all line.
Vector bundles on curves and padic hodge theory request pdf. Degenerating vector bundles in padic hodge theory david hansen. Parallel transport for vector bundles on padic varieties. One of the main results of ff is a complete classi cation of vector bundles on x. Seminar vector bundles on curves and padic hodge theory. Line bundles on rigid varieties and hodge symmetry pdf arxiv, with s.
A is the ring denoted a inf in p adic hodge theory. On certain extensions of vector bundles in padic geometry. For curves or small a ne space over p adic eld, faltings constructed an equivalence between the category of higgs bundles the chern classes may not vanish and \generalised representations using p adic hodge theory and almost etale. In fact, this is the main motivating problem for our work, as it naturally arises in the study of geometric objects such as the stack of vector bundles on xand the ag varieties. A at g bundle on x is a trivializable principal g bundle e. In particular, fargues and fontaine proved that any bundle is determined up to isomorphism by. For this talk some knowledge on p divisible groups is desirable, as these modi. On certain extensions of vector bundles in p adic geometry 3 theorem 1. We discuss recent developments in padic geometry, ranging from foundational.
Then there exists a short exact sequence of vector bundles on x 0. Degenerating vector bundles in p adic hodge theory pdf j. Christopher deninger andannette werner constructed a functor that associates representations of the algebraic fundamental group of an algebraic curve to a class of vector bundles on that curve. The applications to p adic hodge theory, the theorem. A higgs bundle on c is a pair e, where e is a rank n vector bundle on c and. Xsuch that picu 0 and xnu f1gcan be characterized by the following data.
Degenerating vector bundles in p adic hodge theory david hansen. In particular, these vector bundles give rise to representations of the fundamental group. D to the category of continuous representations of the etale fundamental groupoid 1u on cp vector spaces. On p adic lfunctions for hilbert modular forms pdf arxiv, with j.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. By analogy with the p adic case, a geometric theory of minimal ktypes ought to detect both whether a. Din more technical terms, we construct an exact functor. This is a revised version of the lecture notes for the course on p adic geometry given by p. Crew, universal extensions and p adic periods of elliptic curves, comp. Vector bundles on padic curves and parallel transport ii. We define and study a certain category of vector bundles on a p adic curve to which we can associate in a functorial way finite dimensional p adic representations of the geometric fundamental group. Perffp denote the fibered category whose fiber over. We also consider vector bundles of higher rank on acp a. At a few points, we have expanded slightly on the material, in particular so as to provide a full construction of local shimura varieties and general moduli spaces of shtukas, along with some applications to rapoportzink spaces, but otherwise we have tried to keep. Introduction in this note we study the deformation properties of algebraic cycle classes modulo rational equivalence. Vector bundles and padic galois representations imjprg. Degenerating vector bundles in p adic hodge theory pdf. The rst step of p adic analogue of simpson correspondence is given by faltings in fal05.
The curve has already several applications in arithmetic geometry. In the context of the p adic langlands program, it is more natural to consider the mod pof a unitary banach representation than the mod pof a classical admissible representation of a p adic group. The classi cation of vector bundles on the curve gives a short proof of fontaines \weakly admissible admissibleconjecture rst proven by colmezfontaine, which establishes an equi. Werner, vector bundles and p adic representations i, preprint, 2003. We compare this to a construction byfaltings for mumford curves that associates representations of the schottky group to semistable vector bundles of degree 0. Unfortunately we do not have a p adic banach version of the weil representation on a p adic vector space, and this approach cannot be taken. Representations attached to vector bundles on arxiv. Both individuals and organizations that work with arxivlabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Let c be a smooth projective curve of genus g and k k c the canonical bundle. X, which is a k vector space equipped with the hodge. For curves or small a ne space over p adic eld, faltings constructed an equivalence between the category of higgs bundles the chern classes may not vanish and \generalised representations using p adic. The set of such points may well be called the potentially good reduction locus of e.
E x is such a bundle we can choose a ginvariant riemannian metric on e and consider the unitsphere bundle s e x. In 29, the main result was the classification of characteristic subsets. August 22, 2018 abstract we compute the closure relations among the individual hardernarasimhan strata in the moduli stack of rank n vector bundles on the farguesfontaine curve. The category of such objects is one possible p adic. Parallel transport for vector bundles on p adic varieties. Pdf representations attached to vector bundles on curves. Berger has shown that the category of p adic representations of gqp embeds into the category of bpairs by the simple rule v 7. There is a classical relation between the p adic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers, for a variety in characteristic p. All bundles of degree zero whose reduction is strongly semistable belong to this class.
Furthermore, connections on the vector bundles and horizontal sections are missing. Kequivariant vector bundles over x, we recover classical p adic hodge theory as a special case. There are semistable vector bundles on xwith noninteger slopes in contrast to p1. At a few points, we have expanded slightly on the material, in particular so as to provide a full construction of local shimura varieties and general moduli spaces of shtukas. On a compact riemann surface every finite dimensional complex representation of the fundamental group gives rise to a flat vector bundle and hence to a holomorphic vector bundle. If k c, then the classical hodge comparison theorem asserts that h xc. Request pdf vector bundles on curves and p adic hodge theory introduction this text is an introduction to our work 12 on curves and vector bundles in p adic hodge theory. For a vector bundle e on a model of a smooth projective curve over a p adic number field a p adic representation of the geometric fundamental group of x has been defined in work with annette. Tensorproductsin padichodgetheory ucla mathematics. S 2 perffp is the groupoid of rank n vector bundles on s.
Hte on pmin where eare equivariant vector bundles on x. Now, vector bundles on a general curve xwith an open a ne subset u specb. Then there exists a short exact sequence of vector bundles. May 14, 2020 arxivlabs is a framework that allows collaborators to develop and share new arxiv features directly on our website. Ag 17 mar 2009 curves over finite and p adic fields, a comparison christopher deninger 1 the comparison in dw2 and dw4 a partial analogue of the classical narasimhanseshadri correspondence between vector bundles and representations of the fundamental group was developed. We con sider g vector bundles over finite gcwcomplexes. In lectures 1215, we study shtukas with one leg, and their connection to p divisible groups and p adic. Primes for a p adic, convex hull acting canonically on a globally subcommutative vector q. E, and fbe vector bundles on xsuch that the maximum slope in hnd is less than the minimum slope in hnf. We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p adic. The applications to padic hodge theory, the theorem.
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