General Description 

As it is written above on the main page, my interests in general concern number theory and related topics, such as algebraic and arithmetic geometry. The main objects I am working with are global fields in positive characteristic. The most useful tool in my research is the class-field theory. In practice, I usually use the Magma computational algebra system.

Dutch Mathematical Congress 2019, Veldhoven.

In my bachelor thesis I investigated some asymptotic properties of distribution of curves and abelian varieties over finite fields with fixed restriction on the number of points on such curves on the ground field. In my master thesis I constructed curves with many points over some small finite fields, by looking at all unramified abelian coverings of genus three curves. An extended version of  this thesis is available on arXiv. Both my bachelor and master thesis were written under the scientific direction of professor Alexey Zykin. Also, I would like to mention professor Alexey Zaytsev, who was my co-advisor for the master thesis and actually suggested the topic.

My current research is about Anabelian properties of global fields. The main task of Grothendieck’s anabelian geometry is to find relations between an object X and its algebraic fundamental group π(X), provided this group is «sufficiently non-abelian». Suppose X=K is a number or a global function field. In this settings it is natural to consider π(X) as absolute Galois group G = Gal( K^{sep}/K). As pro-finite group the Galois-group G inherits a natural topology and hence it becomes a topological group. It is celebrated theorem due to Neukrich and Uchida which states that in this case isomorphism as topological groups of absolute Galois groups implies isomorphism of fields themselves. The same time, there exists an example which shows that it is not enough to consider abelianization of absolute Galois group to recover isomorphism class of K. In my research I am trying to find sufficient supplementary conditions on isomorphism of abelianizations, such that by using those conditions one indeed has implication about isomorphism of fields. A very related topic is arithmetical equivalence for global fields.

Publications:

  • Notes on Explicit Constructions of Arithmetically Equivalent Global Function Fields via Torsion Points On Drinfeld Modules — this pre-print is available on arXiv.
  • A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem — this pre-print is available on arXiv.
  • A remark On Abelianized Absolute Galois Group of Imaginary Quadratic Fields(Joint work with professor Bart de Smit) — this pre-print is available on arXiv.
  • On Abelianized Absolute Galois Group of Global Function Fields(Joint work with professor Bart de Smit) — this pre-print is available on arXiv. Slides from my talk given at HSE(Moscow) are available here.
  • L-functions and Gassmann Equivalence for Global Function Fields — this pre-print is available on arXiv. Slides from my talk given at IUM(Moscow) are available here.
  • L-functions of genus two abelian coverings of elliptic curves over finite fields — this pre-print is available on arXiv.  Slides from my talk given at IUM(Moscow) and Bordeaux are available here.
  • Curves with many points over finite fields: the class field theory approach —  this pre-print is  available on arXiv

Organized seminars:

  • Drinfeld Modules. In collaboration with Maxim Mornev(Leiden), during the fall 2015 semester I organized research seminar dedicated to Drinfeld Modules.