Related papers: Integral $p$-adic Hodge theory - announcement

In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We…

Let $p$ be a prime number, let $\mathcal{O}_F$ be the ring of integers of a finite field extension $F$ of $\mathbb{Q}_p$ and let $\mathcal{O}_K$ be a complete valuation ring of rank $1$ and mixed characteristic $(0,p)$. We introduce and…

We show that any effective Hodge structure of CM-type occurs (without having to take a Tate twist) in the cohomology of some CM abelian variety over C. As a consequence we get a simple proof of the theorem (due to Hazama) that the usual…

We study several rigidity properties of $p$-adic local systems on a smooth rigid analytic space $X$ over a $p$-adic field. We prove that the monodromy of the log isocrystal attached to a $p$-adic local system is ''rigid'' along irreducible…

Hodge Theory of $p$-adic analytic varieties was initiated by Tate in his 1967 paper on $p$-divisible groups, where he conjectured the existence of a Hodge-like decomposition for the $p$-adic \'etale cohomology of proper analytic varieties.…

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the…

En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des…

In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial…

We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is `effective' for $q=2$; in this case, by…

Let p be a prime. A p-adic functional on a torsion-free abelian group G is a group homomorphism from G to the p-adic integers. The group of all such p-adic functionals is viewed as a p-adic dual group of G, and is studied from the point of…

We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over…

Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends…

For finite dimensional free $C_p$-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of $S^0$ is used to prove the non-existence of certain $C_p$-maps. These are related to Borsuk-Ulam type theorems, and…

We study properties of compactly supported $p$-adic pro-\'etale cohomology of smooth partially proper rigid analytic varieties. In particular, we prove a comparison theorem, in a stable range, with compactly supported syntomic cohomology,…

In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded…

We give a new proof of Ohta's Lambda-adic Eichler-Shimura isomorphism using p-adic Hodge theory and the results of Bloch-Kato and Hyodo on p-adic etale cohomology.

We prove the existence of noncrossed product and indecomposable division algebras over the function field of a smooth p-adic curve, especially when the curve does not admit a smooth model over Z_p. Thus we generalize arXiv 0907.0670. To…

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian…

As an example of relative p-adic Hodge theory, we sketch the construction of the universal admissible filtration of an isocrystal (\phi$-module) over the completion of the maximal unramified extension of Q_p, together with the associated…

Let $\frakX$ be a smooth $p$-adic formal scheme over $\calO_K$ with adic generic fiber $X$. We obtain a global equivalence between the category $\Vect((\frakX)_{\Prism},\overline\calO_{\Prism}[\frac{1}{p}])$ of rational Hodge--Tate crystals…