Ivan Fesenko

Education and professional years

Ivan Fesenko was the winner of an All-Russian mathematical olympiad in 1979. He was an undergraduate student in St Petersburg University and defended his PhD thesis on explicit class field theory in 1986. In 1979-1987 he was involved in conducting various forms of extra-curriculum activities in mathematics for high school students in St Petersburg, such as mathematical circles, mathematical schools and St Petersburg mathematical olympiads. On one occasion he brought a team of students of St Peterburg university, which included Grigorij Perelman, to participate in a mathematical competition with student teams of other universities. He worked at St Petersburg University since 1986. He was awarded the Prize of the Petersburg Mathematical Society in 1992. Since 1995 he is professor in pure mathematics at University of Nottingham.

Ivan Fesenko contributed to number theory and to several intra-disciplinary developments in related areas, while his students contributed to algebraic geometry, representation theory, model theory. Since 2015 he is the principal investigator of a research team of Universities of Nottingham and Oxford supported by EPSRC Programme Grant on Symmetries and Correspondences and intra-disciplinary developments.

Work in number theory, class field theory and higher class field theory

In his first papers Fesenko discovered several types of explicit formulas for the generalized Hilbert symbol on local fields and higher local fields in the case of even residue characteristic. There formulas belong to the branch of Vostokov's explicit formulas. He developed several explicit generalizations of class field theory, using different methods. He extended the method of Neukirch in class field theory to deal with generalized class formations that do not satisfy the property of Galois descent, and applied the extended method in higher class field theory. Together with Kato, he is the main contributor to higher local class field theory. This theory uses quotients of Milnor K-groups instead of the multiplicative group of a usual local field with finite residue field.

Fesenko developed other types of class field theories. He constructed p-class field theory for local fields with perfect and imperfect residue field where indices of norm groups can be infinite. In 2000 he introduced "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields and proved its main theorems. This arithmetic theory can be viewed as an alternative to the representation theoretical local Langlands correspondence. He established an appropriate version of inverse theorem to the Hasse-Arf theorem, contributed to ramification theory of infinite extension and ramification theory for higher local fields.

Fesenko is a coauthor of a textbook on local fields and a coeditor of a volume on higher local fields..

Work on higher Haar measure and higher harmonic analysis

Solving a long-standing problem, Fesenko introduced a higher Haar measure and integration on various higher local and adelic objects and proved main theorems of this higher harmonic analysis. This measure takes values in the formal power series over complex numbers, while in dimension one the theory is the usual Haar measure. The measure is translation invariant, finitely additive and countably additive in some refined sense. In the case of formal power series over reals or complex numbers, the associated integral has various links with the Feynman path integral.

Work on higher adelic structures

Fesenko discovered a new adelic structure on relative surfaces, which he called an analytic adelic structure. The relation between the usual, geometric adelic structure and the analytic adelic structure is an adelic version of the relation between the ring of integers of rank one and the ring of integers of rank two of a two-dimensional non-archimedean local field. Solving a long-standing problem and further developing the theory of geometric adelic structrures, Fesenko obtained the first full two-dimensional adelic proof of the Riemann-Roch theorem for surfaces over perfect fields whose predecessor for curves is the proof of Artin and Iwasawa.

Work on arithmetic zeta functions, FIT theory and three further developments

Fesenko pioneered the study of zeta functions in higher dimensions by developing his theory of higher adelic zeta integrals. These integrals are defined using the higher Haar measure and ojects coming from higher class field theory. The zeta integral of a surface is closely related to the square of the zeta function of the surface. Using the higher adelic zeta integral Fesenko generalized Iwasawa-Tate theory and Tate's thesis from 1-dimensional global fields to 2-dimensional objects such as proper regular models of elliptic curves over global fields. Via higher adelic duality and higher harmonic analysis, FIT theory reduces the study of the zeta function of the surface to the study of a certain adelic boundary integral and its properties. Thus, the investigation of the adelic boundary integral becomes the key activity in the FIT approach to arithmetic of elliptic curves. Three developments followed the main theorem of FIT theory.

The first development is the study of functional equation and meromorphic continuation of the Hasse zeta function of a proper regular model of an elliptic curve over a global field. This study led Fesenko to introduce his new mean-periodicity correspondence between the arithmetic zeta functions and mean-periodic elements of the space of smooth functions on the real line of not more than exponential growth at infinity. This correspondence can be viewed as a weaker version of Langlands correspondence where L-functions and replaced by zeta functions and automorphicity is replaced by mean-periodicity.. This work was followed by joint work with Suzuki and Ricotta.

The second development is an application to the generalized Riemann hypothesis, which in this higher theory is reduced to a certain positivity property of small derivatives of the boundary function and to the properties of the spectrum of the Laplace transform of the boundary function.

The third development is higher adelic study of relations between the arithmetic and analytic ranks of an elliptic curve over a global field, which in conjectural form are stated in the Birch and Swinnerton-Dyer conjecture for the zeta function of elliptic surfaces. This new method uses FIT theory, two adelic structures: the geometric additive adelic structure and the arithmetic multiplicative adelic structure and an interplay between them motivated by higher class field theory. The two adelic structures have some similarity to two symmetries in inter-universal Teichmüller theory of Mochizuki.

Other contributions

For his study of infinite ramification theory, Fesenko introduced a closed torsion free hereditarily just infinite subgroup of the Nottingham group. This group is now called the Fesenko group.

Ivan Fesenko has worked on and played an active role in organizing the study of inter-universal Teichmüller theory of Shinichi Mochizuki. He is the author of a survey and a general article on this theory. He co-organized two international workshops on IUT.