Annette Werner

Vector bundles on p-adic curves and parallel transport

Abstract We define functorial isomorphisms of parallel transport along etale 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.

Bruhat-Tits buildings and analytic geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group.

Line Bundles and p -Adic Characters

For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.

Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces

This paper generalizes Manins approach towards a geometrical interpretation of Arakelov theory at infinity to linear cycles on projective spaces. We show how to interpret certain non-Archimedean Arakelov intersection numbers of linear cycles on

Analytification and Tropicalization Over Non-archimedean Fields

P^{n-1}

Automorphisms of Drinfeld half-spaces over a finite field

with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n). This geometric setting has an Archimedean analogue, namely the Riemannian symmetric space associated to SL(n,C), which we use to interpret analogous Archimedean intersection numbers of linear cycles in a similar way.

A tropical view on Bruhat-Tits buildings and their compactifications

In this paper, we provide an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker et al. (Algebr. Geom. 3, 63–105 (2016); Contemporary Mathematics, vol 605, pp 93–121. American Mathematical Society, Providence, 2013) in the case of curves, we explain a result from Cueto et al. (Math. Ann. 360, 391–437 (2014)) comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of most of the results in Gubler et al. (Adv. Math. 294, 150–215 (2016)), where a general higher-dimensional theory is developed. In particular, we explain the construction of generalized skeleta in Gubler et al. (Adv. Math. 294, 150–215 (2016)) which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.

Local heights on abelian varieties with split multiplicative reduction

We show that the automorphism group of Drinfelds half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

Non-Archimedean Geometry and Applications

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.

Non-Archimedean Analytic Geometry

We express Néron functions and Schneiders local p-adic height pairing on an abelian variety A with split multiplicative reduction with theta functions and their automorphy factors on the rigid analytic torus uniformizing A.Moreover, we show formulas for the ρ-splittingsof the Poincaré biextension corresponding to Nérons and Schneiders local height pairings.