Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe $p$-adic Gelfand-Kapranov-Zelevinsky hypergeometric complex
160
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Neron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.
We use Scholzes framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the etale, pro-etale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $mathbb{C}_p$ this enables us to prove a category equivalence between $mathbb{C}_p$-local systems with integral models, and modules under the $v$-structure sheaf which modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. 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.
Given three arbitrary vector bundles on the Fargues-Fontaine curve where one of them is assumed to be semistable, we give an explicit and complete criterion in terms of Harder-Narasimha polygons on whether there exists a short exact sequence among them. Our argument is based on a dimension analysis of certain moduli spaces of bundle maps and bundle extensions using Scholzes theory of diamonds.
We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
