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Veronika Ertl

Publications and source records attributed to Veronika Ertl.

15 recordsLinked to original sources
arXiv2025

Comparison of different Tate conjectures

For an abelian variety $A$ over a finitely generated field $K$ of characteristic $p > 0$, we prove that the algebraic rank of $A$ is at most a suitably defined analytic rank. Moreover, we prove that equality, i.e., the BSD rank conjecture, holds for $A/K$ if and only if a suitably defined Tate--Shafarevich group of $A/K$ (1) has finite $\ell$-primary component for some/all $\ell \neq p$, or (2) finite prime-to-$p$ part, or (3) has $p$-primary part of finite exponent, or (4) is of finite exponent. There is an algorithm to verify those conditions for concretely given $A/K$.

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arXiv2025

Berthelot's conjecture via homotopy theory

We use motivic methods to give a quick proof of Berthelot's conjecture stating that the push-forward map in rigid cohomology of the structural sheaf along a smooth and proper map has a canonical structure of overconvergent F-isocrystal on the base.

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arXiv2025

Rigid analytic reconstruction of Hyodo--Kato theory

We give a new and very intuitive construction of Hyodo--Kato cohomology and the Hyodo--Kato map, based on logarithmic rigid cohomology. We show that it is independent of the choice of a uniformiser and study its dependence on the choice of a branch of the $p$-adic logarithm. Moreover, we show the compatibility with the classical construction of Hyodo--Kato cohomology and the Hyodo--Kato map.)

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arXiv2025

Integral p-adic cohomology theories for open and singular varieties

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically étale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.

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arXiv2024

On the $v$-Picard group of Stein spaces

We study the image of the Hodge-Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $Z_p$-module that maps, via the Bloch-Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit discs of dimension strictly greater than $1$.

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arXiv2024

Poincaré duality for rigid analytic Hyodo--Kato theory

The purpose of this paper is to establish Hyodo--Kato theory with compact support for semistable schemes through rigid analytic methods. To this end we introduce several types of log rigid cohomology with compact support. moreover we show that additional structures on the (rigid) Hyodo--Kato cohomology and the Hyodo--Kato map introduced in our previous paper are compatible with Poincaré duality. Compared to the crystalline approach, the constructions are explicit yet versatile, and hence suitable for computations.

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arXiv2019

Comparison between rigid syntomic and crystalline syntomic cohomology for strictly semistable log schemes with boundary

We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p). In case a good compactification exists, we compare this cohomology theory to Nekovář-Nizioł's crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Große-Klönne's rigid Hyodo-Kato theory and a generalisation of it for strictly semistable log schemes with boundary.

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arXiv2019

A new proof of a vanishing result due to Berthelot, Esnault, and Rülling

The goal of this small note is to give a more concise proof of a result due to Berthelot, Esnault, and Rülling. For a regular, proper, and flat scheme $X$ over a discrete valuation ring of mixed characteristic $(0,p)$, it relates the vanishing of the cohomology of the structure sheaf of the generic fibre of $X$ with the vanishing of the Witt vector cohomology of its special fibre. We use as a critical ingredient results and constructions by Beilinson and Nekovář--Nizioł related to the $h$-topos over a $p$-adic field.

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arXiv2018

Witt differentials in the h-topology

Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.

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arXiv2017

Integral Comparison of Monsky-Washnitzer and overconvergent de Rham-Witt cohomology

The goal of this small note is to extend a result by Christopher Davis and David Zureick-Brown on the comparison between integral Monsky-Washnitzer cohomology and overconvergent de~Rham-Witt cohomology for a smooth variety over a perfect field of positive characteristic $p$ to all cohomological degrees independent of the dimension of the base or the prime number $p$. Le but de ce travail est de prolonger un résultat de Christopher Davis et David Zureick-Brown concernant la comparaison entre la cohomologie de Monsky-Washnitzer entière et la cohomologie de de~Rham-Witt surconvergente d'une variété lisse sur un coprs parfait de charactéristique positive $p$ à tous les degrés cohomologiques indépnedent de la dimension de base et du nombre premier $p$.

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arXiv2016

Syntomic cohomology and $p$-adic motivic cohomology

We prove a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser from small Tate twists to all twists and uses as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently in Colmez-Niziol.

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arXiv2016

Comparison between Rigid and Overconvergent Cohomology with Coefficients

For a smooth scheme over a perfect field of characteristic p>0, we generalise a definition of Bloch and introduce overconvergent de Rham-Witt connections. This provides a tool to extend the comparison morphisms of Davis, Langer and Zink between overconvergent de Rham-Witt cohomology and Monsky-Washnitzer respectively rigid cohomology to coefficients.

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arXiv2016

Full faithfulness for overconvergent F-de Rham-Witt connections

Let X be a smooth variety over a perfect field of characteristic p>0. In this small note we define overconvergent F-de Rham-Witt connections as an analogue for F-crystals over proper schemes. We prove that the forgetful functor from the category of overconvergent F-de Rham-Witt connections to the category of convergent F-de Rham-Witt connections is fully faithful. The argument is an analogue of a discussion by Kedlaya on full faithfulness of overconvergent F-isocrystals.}

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arXiv2014

Overconvergent Chern Classes and Higher Cycle Classes

The goal of this work is to construct integral Chern classes and higher cycle classes for a smooth variety over a perfect field of characteristic p>0 that are compatible with the rigid Chern classes defined by Petrequin. The Chern classes we define have coefficients in the overconvergent de Rham-Witt complex of Davis, Langer and Zink and the construction is based on the theory of cycle modules discussed by Rost. We prove a comparison theorem in the case of a quasi-projective variety.

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