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Thomas H. Geisser

Publications and source records attributed to Thomas H. Geisser.

16 recordsLinked to original sources
arXiv2026

On Frobenius rigidity for motivic cohomology

We study how motivic and étale motivic cohomology of a smooth and proper variety $X$ changes under extensions of algebraically closed base fields $k$. We show that with finite coefficients, they are independent of $k$ away from the characteristic. At the characteristic, étale cohomology does depend on $k$, whereas for motivic cohomology this is only known in weights $0,1,\dim X$. We then consider the fiber of Frobenius on motivic cohomology and étale motivic cohomology for varieties defined over finite fields, i.e., Weil-étale cohomology. Combining the results of the first part with the structure theory of perfect unipotent group schemes, we show that this fiber is independent of $k$ with finite coefficients. Finally, we give some results and examples with integral coefficients.

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arXiv2024

Brauer and Neron-Severi groups of surfaces over finite fields

We give a version of the Artin-Tate formula for surfaces over finite fields not assuming Tate's conjecture. It gives an equality between terms related to the Brauer group on the one hand and terms related to the Neron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to gives sharp estimates on the size of the Brauer group of abelian surfaces depending on the p-rank.

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arXiv2022

On Integral Class field theory for varieties over $p$-adic fields

Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$ its generic fiber. We show, under some assumptions on $\mathcal X_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}(\mathcal X_K, \mathbb Z(d)) \simeq π_1^{ab}(\mathcal X_K)_{W}$ from a new cohomology theory to an integral model $π_1^{ab}(\mathcal X_K)_{W}$ of the abelianized geometric fundamental groups $π_1^{ab}(\mathcal X_K)^{geo}$. After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups.

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arXiv2022

Special values of L-functions of one-motives over function fields

The purpose of this paper is to give a formula for the leading coefficient at $s=1$ of the $L$-function of one-motives over function fields in terms of Weil-étale cohomology, generalizing the Weil-étale version of the Birch and Swinnerton-Dyer conjecture in the authors' previous work. As a consequence we express the Tamagawa number of a torus introduced by Ono-Oesterlé in terms of Weil-étale cohomology, and reprove their Tamagawa number formula.

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arXiv2021

Pontryagin duality for varieties over $p$-adic fields

We define cohomological complexes of locally compact abelian groups associated with varieties over $p$-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

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arXiv2021

On the kernel of the Brauer-Manin pairing

Let $\mathcal X$ be a regular scheme, flat and proper over the ring of integers of a $p$-adic field, with generic fiber $X$ and special fiber $\mathcal X_s$. We study the left kernel $Br(\mathcal X)$ of the Brauer-Manin pairing $Br(X)\times CH_0(X)\to \mathbb Q/\mathbb Z$. Our main result is that the kernel of the reduction map $Br(\mathcal X)\to Br(\mathcal X_s)$ is the direct sum of $(\mathbb Q/\mathbb Z[\frac{1}{p}])^s\oplus (\mathbb Q/\mathbb Z)^t$ and a finite $p$-group, where $s+t= ρ_{\mathcal X_s}-ρ_X-I+1$, for $ρ_{\mathcal X_s}$ and $ρ_X$ the Picard numbers of $\mathcal X_s$ and $X$, and $I$ the number of irreducible components of $\mathcal X_s$. Moreover, we show that $t>0$ implies $s>0$.

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arXiv2018

Tate's conjecture and the Tate-Shafarevich group over global function fields

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the formula given for a surface.

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arXiv2017

Duality for integral motivic cohomology

We discuss duality pairings on integral étale motivic cohomology groups of regular and proper schemes over algebraically closed fields, local fields, finite fields, and arithmetic schemes.

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arXiv2016

On the structure of etale motivic cohomology

We discuss the structure of integral etale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes.

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arXiv2009

Suslin's singular homology and cohomology

We discuss Suslin's singular homology and cohomology. In the first half we examine the p-part in characteristic p, and the situation over non-algebraically closed fields. In the second half we focus on finite base fields. We study finite generation properties, and give a modified definition which behaves like a homology theory: in degree zero it is a copy of Z for each connected component, in degree one it is related to the abelianized (tame) fundamental group, even for singular schemes, and it is expected to be finitely generated in general.

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arXiv2005

Arithmetic cohomology over finite fields and special values of zeta-functions

We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.

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arXiv2004

Weil-etale cohomology over finite fields

We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale cohomology. In the second half of the paper, we apply this to study the Weil-etale cohomology of the motivic complex for smooth and projective varieties. These groups are expected to be finitely generated, to give an integral model for l-adic cohomology, and to be related to special values of the zeta function. We give necessary and sufficient conditions for this to hold, and examples.

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arXiv2002

Weil-etale motivic cohomology

We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence, we get a long exact sequence relating Weil-etale cohomology to etale cohomology, show that for finite coefficients the cohomology theories agree, and with rational coefficients a Weil-etale cohomology group is the direct sum of two etale cohomology groups. In the second half of the paper we restrict ourselves to Weil-etale cohomology of the motivic complex. We show that for smooth projective varieties over finite fields, finite generation of Weil-etale cohomology is equivalent to Weil-etale cohomology being an integral model of l-adic cohomology, and also equivalent to the conjunction of Tate's conjecture and (rational) equality of rational and numerical equivalence. We give several examples where these conjectures hold, and express special values of zeta functions in terms of Weil-etale cohomology.

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