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.
    PDF file, DOI 10.4171/EMSS/10, arXiv:1408.1441 (math.NT)

    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 and duality for arithmetic schemes in negative weights,
    December 2020 (revised November 2021), submitted for publication.
    PDF file, arXiv:2012.11034 (math.AG, math.NT), LaTeX source (GitHub)

    Abstract: 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 assumes finite generation of suitable étale motivic cohomology groups.

  3. Alexey Beshenov, Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers,
    February 2021 (revised November 2021), submitted for publication.
    PDF file, arXiv:2102.12114 (math.AG, math.NT), LaTeX source (GitHub)

    Abstract: 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 \(\zeta (X,s)\) 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 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 \(X\).

  4. Alexey Beshenov, Zeta-values of one-dimensional arithmetic schemes at strictly negative integers,
    November 2021, submitted for publication.
    PDF file, arXiv:2111.13398 (math.AG, math.NT), LaTeX source (GitHub)

    Abstract: 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 \(\zeta (X,s)\), 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 \(\eta \in X\) with \(\operatorname{char} \kappa (\eta) = 0\), the extension \(\kappa (\eta)/\mathbb{Q}\) 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.