Ivan Fesenko

Ivan Fesenko
Born St Petersburg, Russia
Fields Mathematician
Institutions University of Nottingham
Alma mater Saint Petersburg State University
Known for number theory, class field theory, higher class field theory, zeta functions and integrals, adelic analysis and geometry, Fesenko groups
Notable awards Petersburg Mathematical Society Prize (1992)

Ivan Borisovich Fesenko (Russian: Иван Борисович Фесенко; born 1962) is a mathematician working in number theory and other areas of mathematics. In 1992 Fesenko won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on class field theory.[1]

Work

Fesenko has worked to extend and generalize several theories for one-dimensional objects in algebraic number theory to a higher-dimensional version for arithmetic schemes.

In class field theory Fesenko constructed an explicit class field theory for complete objects associated to arithmetic schemes,[2][3] which is part of higher class field theory where Milnor K-groups of the fields play a central role. He developed an explicit class field theory for local fields with perfect and imperfect residue field.[4][5] Fesenko initiated a "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields[6] which relates quotients of the field of norms with the Galois group via a 1-cocycle. He is a coauthor of a textbook on local fields[7] and a coeditor of a volume on higher local fields.[8]

Generalizing the Haar measure and integration to non locally compact objects associated to arithmetic schemes, Fesenko developed a translation invariant measure, integration and Fourier transform on higher-dimensional local fields.[9] Extending the theory of geometric adele rings associated to arithmetic surfaces he introduced analytic adelic objects associated to rank two integral structures and developed the theory of measure and integration on them.[10]

Fesenko initiated the study of zeta functions in higher dimensions using zeta integrals. He introduced zeta integrals on arithmetic schemes of dimension two, generalized Tate's thesis and proved a two-dimensional version which reduces the study of the zeta function to the study of geometric and analytic properties of adelic spaces and integrals over them.[11] His work relates adelic dualities and measure theoretical and topological properties of quotients of adelic spaces with properties of the arithmetic zeta functions.

As one of applications and developments of his work, a new mean-periodicity correspondence was proposed as a weaker version of Langlands correspondence. It relates 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. Unlike Langlands correspondence, this correspondence is of commutative nature. [12][13]

Other applications include a new approach to the generalized Riemann hypothesis for the zeta function of elliptic surfaces and to the Birch and Swinnerton-Dyer conjecture about special values of the zeta function. For the latter the zeta integral and explicit higher class field theory provide a direct relation between the arithmetic and geometric ranks.

References

  1. ↑ "Young mathematician prize of the Petersburg Mathematical Society".
  2. ↑ I. Fesenko (1992). "Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic". St. Petersburg Mathematical Journal, vol. 3. pp. 649–678.
  3. ↑ Fesenko, I. (1995). "Abelian local p-class field theory". Math. Ann. 301: 561–586. doi:10.1007/bf01446646.
  4. ↑ I. Fesenko (1994). "Local class field theory: perfect residue field case". Russ. Acad. Scienc. Izvest. Math., vol. 43. pp. 65–81.
  5. ↑ Fesenko, I. (1996). "On general local reciprocity maps". Journal fur die reine und angewandte Mathematik 473: 207–222.
  6. ↑ Fesenko, I. (2001). "Nonabelian local reciprocity maps". Class Field Theory – Its Centenary and Prospect, Advanced Studies in Pure Math.: 63–78. ISBN 4-931469-11-6.
  7. ↑ Fesenko, I.B.; Vostokov, S.V. (2002). Local Fields and Their Extensions, Second Revised Edition, Amer. Math. Soc. ISBN 978-0-8218-3259-2.
  8. ↑ I. Fesenko and M. Kurihara (2000). "Invitation to higher local fields, Geometry and Topology Monographs, ISSN 1464-8997". Geometry and Topology Publications.
  9. ↑ I. Fesenko (2003). "Analysis on arithmetic schemes. I". Documenta Mathematica. pp. 261–284. ISBN 978-3-936609-21-9.
  10. ↑ Fesenko, I. (2008). "Adelic study of the zeta function of arithmetic schemes in dimension two". Moscow Mathematical Journal 8: 273–317.
  11. ↑ I. Fesenko (2010). "Analysis on arithmetic schemes. II". Journal of K-theory, vol. 5. pp. 437–557.
  12. ↑ I. Fesenko (2010). "Analysis on arithmetic schemes. II". Journal of K-theory, vol. 5. pp. 437–557.
  13. ↑ Fesenko, I.; Ricotta, G.; Suzuki, M. (2012). "Mean-periodicity and zeta functions". Ann. L'Inst. Fourier 62: 1819–1887. doi:10.5802/aif.2737.

External links