We define functorial isomorphisms of parallel transport alongétale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.MSC (2000) 14H60, 14H30, 11G20 IntroductionOn 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. By a theorem of Weil, one obtains precisely the holomorphic bundles whose indecomposable components have degree zero [W]. It was proved by Narasimhan and Seshadri [Na-Se] that unitary representations give rise to polystable bundles of degree zero. Moreover, every stable bundle of degree zero comes from an irreducible unitary representation.The present paper establishes a partial p-adic analogue of this theory, generalized to representations of the fundamental groupoid. The following is our main result.Recall that a vector bundle on a smooth projective curve over a field of characteristic p is called strongly semistable if the pullbacks of E by all non-negative powers of the absolute Frobenius morphism are semistable. Let X be a smooth projective curve over Q p and let o be the ring of integers in C p . A model X of X is a finitely presented flat and proper scheme over Z p with generic fibre X. The special fibre X k is then a union of projective curves over k = F p . We say that a vector bundle E on X Cp = X ⊗ C p has strongly semistable reduction of degree zero if the following is true: E can be extended to a vector bundle E on X o = X ⊗ o for some model X of X such that the pullback of the special fibre E k of E to the normalization of each irreducible component of X k is strongly semistable of degree zero. We say that 1 E has potentially strongly semistable reduction of degree zero if there is a finité etale morphism α : Y → X of smooth projective curves such that α * E has strongly semistable reduction of degree zero.Theorem Let E be a vector bundle on X Cp with potentially strongly semistable reduction of degree zero. Then there are functorial isomorphisms of "parallel transport" alongétale paths between the fibres of E Cp on X Cp . In particular one obtains a representation ρ E,x of π 1 (X, x) on E x for every point x in X(C p ). The parallel transport is compatible with tensor products, duals, internal homs, pullbacks and Galois conjugation.The theorem applies in particular to line bundles of degree zero on X Cp . In this case the p-part of the corresponding character of π 1 (X, x) was already constructed by Tate using Cartier duality for the p-divisible group of the abelian scheme Pic 0 . His method does not extend to bundles of higher rank.Let us now discuss the contents of the paper in more detail. Afterwards we can sketch the proof of the theorem...

Achromatopsia linked to variations in the CNGA3 gene is associated with day blindness, poor visual acuity, photophobia, and involuntary eye movements owing to lack of cone photoreceptor function. No treatment is currently available. OBJECTIVE To assess safety and vision outcomes of supplemental gene therapy with adeno-associated virus (AAV) encoding CNGA3 (AAV8.CNGA3) in patients with CNGA3-linked achromatopsia. DESIGN, SETTING, AND PARTICIPANTS This open-label, exploratory nonrandomized controlled trial tested safety and vision outcomes of gene therapy vector AAV8.CNGA3 administered by subretinal injection at a single center. Nine patients (3 per dose group) with a clinical diagnosis of achromatopsia and confirmed biallelic disease-linked variants in CNGA3 were

ABSTRACT. Let K be a complete, algebraically closed non-archimedean field with ring of integers K • and let X be a K-variety. We associate to the data of a strictly semistable K • -model X of X plus a suitable horizontal divisor H a skeleton S(X , H) in the analytification of X. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker-Payne-Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on S(X , H). For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a wellknown result by Sturmfels-Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G, k) to the Berkovich analytic space G an asscociated with G. Composing this map with the projection of G an to its flag varieties, we define a family of compactifications of B (G, k). This generalizes results by Berkovich in the case of split groups.Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them. AMS classification (2000): 20E42, 51E24, 14L15, 14G22.1. In the mid 60ies, F. Bruhat and J. Tits initiated a theory which led to a deep understanding of reductive algebraic groups over valued fields [BT72], [BT84]. The main tool (and a concise way to express the achievements) of this long-standing work is the notion of a building. Generally speaking, a building is a gluing of (poly)simplicial subcomplexes, all isomorphic to a given tiling naturally acted upon by a Coxeter group [AB08]. The copies of this tiling in the building are called apartments and must satisfy, by definition, strong incidence properties which make the whole space very symmetric. The buildings considered by F. Bruhat and J. Tits are Euclidean ones, meaning that their apartments are Euclidean tilings (in fact, to cover the case of non-discretely valued fields, one has to replace Euclidean tilings by affine spaces acted upon by a Euclidean reflection group with a non-discrete, finite index, translation subgroup [Tit86]). A Euclidean building carries a natural non-positively curved metric, which allows one to classify in a geometric way maximal bounded subgroups in the rational points of a given non-Archimedean semisimple algebraic group. This is only an instance of the strong analogy between the Riemannian symmetric spaces associated with semisimple real Lie groups and Bruhat-Tits buildings [Tit75]. This analogy is our guideline here.Indeed, in this paper we investigate Bruhat-Tits buildings and their compactification by means of analytic geometry over non-Archimedean valued fields, as developed by V. Berkovich -see [Ber98] for a survey. Compactifications of symmetric spaces is now a very classical topic, with well-known applications to group theory (e.g., group cohomology [BS73]) and to number theory (via the study of some relevant moduli spaces modeled on Hermitian symmetric spaces [Del71]). For deeper motivation and a broader scope on compactifications of symmetric spaces, we refer to the recent book [BJ06], in which the case of locally symmetric varieties is also covered. One of our main results is to construct for each semisimple group G over a suitable non-Archimedean valued field k, a family of compactifications of the Bruhat-Tits building B(G, k) of G over k. This fami...

We construct a compactification X of the Bruhat-Tits building X associated to the group P GL(V ) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to X which extends the reduction map from Drinfeld's p-adic symmetric domain to the building X.