We introduce a new approach to computing topological cyclic homology by means of a descent spectral sequence. We carry out the computation for a p-adic local field with Fp-coefficients, including the case p=2 which was only covered by motivic methods except in the totally unramified case.
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the etale cohomology with partial compact support of de Rham $mathbb Z_p$-local systems, and show that they are compatible with Po
incare duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of etale cohomology with partial compact support of any $mathbb Z_p$-local systems, and establish the Poincare duality for such cohomology after inverting $p$.
We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer etale and pro-Kummer etale topology for such spaces. We also establish the primitive comparison theorem in this context, and deduce
from it some related cohomological finiteness or vanishing results.
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic etale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griff
iths transversality, which we view as a $p$-adic analogue of Delignes classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.
We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite extension of Q_p or an algebraically closed nonarchimed
ean field containing Q_p. This result manifests as a special case of a more general finiteness result for the higher direct images of a relative (phi, Gamma)-module along a smooth proper morphism of rigid analytic spaces over a mixed-characterstic nonarchimedean field.
In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholz
es pro-etale topology. In this paper, we first extend Kiehls theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a suitable infinite etale cover; in this case, one can decomplete the period sheaves and establish an analogue of the theorem of Cherbonnier-Colmez on the overconvergence of p-adic Galois representations. As an application, we show that relative (phi, Gamma)-modules in our sense coincide with the relative (phi, Gamma)-modules constructed by Andreatta and Brinon in the geometric setting where the latter can be constructed. As another application, we establish that the category of pseudocoherent (phi, Gamma)-modules on an arbitrary rigid analytic space over a p-adic field is abelian, satisfies the ascending chain condition, and is stable under various natural derived functors (including Hom, tensor product, and pullback). Applications to the etale cohomology of pro-etale local systems will be given in a subsequent paper.
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 f
irst 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.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we pr
ove the non-rationality of the geometric unit root L-functions.
We prove that the eigencurve associated to a definite quaternion algebra over $QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint u
nion of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.
We prove that the Coleman-Mazur eigencurve is proper over the weight space for any prime p and tame level N.
