arXiv subjects

Alexey Beshenov

Publications and source records attributed to Alexey Beshenov.

4 recordsLinked to original sources

arXiv2025

Weil-étale cohomology and duality for arithmetic schemes in negative weights

Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-étale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec} \mathbb{Z}$) and $n \in \mathbb{Z}$. In the case when $n < 0$, we generalize their construction to an arbitrary arithmetic scheme $X$, thus removing the proper and regular assumption. The construction uses étale motivic cohomology groups $H^i(X_\text{ét}, \mathbb{Z}^c(n))$, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for $n < 0$. We give a class of X for which finite generation is known, and hence $H^i_\text{W,c} (X, \mathbb{Z} (n))$ is defined unconditionally.

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arXiv2021

Zeta-values of one-dimensional arithmetic schemes at strictly negative integers

Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $ζ(X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the étale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $η\in X$ with $\operatorname{char} κ(η) = 0$, the extension $κ(η)/\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-étale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-étale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114].

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arXiv2021

Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers

Following the ideas of Flach and Morin (Doc. Math. 23 (2018), 1425--1560), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function $ζ(X, s)$ at $s = n < 0$, where $X$ is a separated scheme of finite type over $\operatorname{Spec} \mathbb{Z}$. We prove that the conjecture is compatible with closed-open decompositions of schemes and with affine bundles, and consequently, that it holds for cellular schemes over certain one-dimensional bases. This is a continuation of arXiv:2012.11034, which gives a construction of Weil-étale cohomology for $n < 0$ under the mentioned assumptions on $X$.

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arXiv2015

Rational points on analytic varieties

These are expanded notes of the mini-courses on Pila's work that Yuri Bilu gave in Basel in April 2011, Yaroslavl in August 2011 and Chennai in February 2012. The topics covered include the Bombieri-Pila theorem, its extensions and applications to some prominent conjectures (Manin-Mumford, André-Oort).

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