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Register a new userThe $p$-adic Corlette-Simpson correspondence for abeloids
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For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $mathbb Q_p$, we construct a $p$-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous $K$-linear representations of the Tate module and a certain subcategory of the Higgs bundles on $A$. To do so, our central object of study is the category of vector bundles for the $v$-topology on the diamond associated to $A$. We prove that any pro-finite-etale $v$-vector bundle can be built from pro-finite-etale $v$-line bundles and unipotent $v$-bundles. To describe the latter, we extend the theory of universal vector extensions to the $v$-topology and use this to generalize a result of Brion by relating unipotent $v$-bundles on abeloids to representations of vector groups.
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We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection over its base change to B_dR, which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties.
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterle) and to the study of random $p$-adic polynomial systems of equations.
For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the invariant cycles theorem with trivial coefficients (see Chiarellotto, 1999) i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber of $X$ coincides with the first rigid cohomology group of its special fiber, without the hypothesis that the residue field of $cal V$ is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of $X$ with coefficients convergent $F$-isocrystals given in Coleman and Iovita (2010). We apply these ideas to the case where the coefficients are unipotent convergent $F$-isocrystals defined on the special fiber (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness.
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmuller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an over-convergent $F$-isocrystal. It is the crystalline companion of the $ell$-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dworks theory and the theory of arithmetic $mathcal D$-modules of Berthelot.
We develop a theory of etale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the etale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
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