Here is my mathematical writing: PhD thesis, publications and preprints. Any feedback is welcome!

## PhD thesis

Here is the original manuscript of my PhD thesis, advised by Baptiste Morin (Université de Bordeaux) and Bas Edixhoven (Universiteit Leiden), defended in Leiden on December 10, 2018.

For a more streamlined exposition and more complete results (also with some corrections of minor misprints), see my preprints arXiv:2012.11034 and arXiv:2102.12114, listed below.

## Publications and preprints

1. Alexey Beshenov, Margaret Bilu, Yuri Bilu, Purusottam Rath, Rational points on analytic varieties, EMS Surv. Math. Sci. 2 (2015), no. 1, 109–130.

Abstract: This is a brief, informal and very elementary introduction into the work of Pila and others about counting rational points on analytic and, more generally, definable sets. We also discuss some of the most spectacular applications.

2. Alexey Beshenov, Weil-étale cohomology for arbitrary arithmetic schemes and $n < 0$. Part I: Construction of Weil-étale complexes, 2020, preprint arXiv:2012.11034, submitted.

Abstract: Flach and Morin constructed in (Doc. Math. 2 (2018), 1425–1560) Weil-étale cohomology $H^i_{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (that is, 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 assumes finite generation of suitable étale motivic cohomology groups.

This is the first part in a series of two papers. In the present text we consider the definition and basic properties of Weil-étale cohomology. The second part will deal with the relation of $H^i_{W,c} (X, \mathbb{Z} (n))$ with the special value of zeta function $\zeta (X, s)$ at $s = n < 0$.

3. Alexey Beshenov, Weil-étale cohomology for arbitrary arithmetic schemes and $n < 0$. Part II: The special value conjecture, 2021, preprint arXiv:2102.12114.

Abstract: Following the ideas of Flach and Morin (Doc. Math. 2 (2018), 1425–1560), we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function $\zeta (X,s)$ 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 affine bundles, and as a consequence, that it holds for cellular schemes over certain 1-dimensional bases.

This is a continuation of author’s preprint arXiv:2012.11034, which gives a construction of Weil-étale cohomology for $n < 0$ under the mentioned assumptions on $X$.

4. Alexey Beshenov, Zeta-values of one-dimensional arithmetic schemes at $n < 0$, 2021, in preparation (comments are welcome).

Abstract: Let $X \to \operatorname{Spec} \mathbb{Z}$ be an arithmetic scheme (separated, of finite type) of Krull dimension $1$. We write down a formula for the special value of $\zeta (X,s)$ at $s = n < 0$, in terms of étale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $\eta \in X$ with char $\kappa (\eta) = 0$, the extension $\kappa(\eta)/\mathbb{Q}$ is abelian. Further, we conjecture that the formula holds for any one-dimensional arithmetic scheme.

This is a consequence of Weil-étale formalism developed by the author in arXiv:2012.11034 and arXiv:2102.12114, following the work of Flach and Morin (Doc. Math. 2 (2018), 1425–1560).

In particular, we calculate Weil-étale cohomology of one-dimensional arithmetic schemes.