RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Funktsional. Anal. i Prilozhen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Funktsional. Anal. i Prilozhen., 2011, Volume 45, Issue 2, Pages 23–44 (Mi faa3037)  

This article is cited in 16 scientific papers (total in 17 papers)

Krichever Formal Groups

V. M. Buchstaber, E. Yu. Bun'kova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: On the basis of the general Weierstrass model of the cubic curve with parameters $\mu=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_6)$, the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in $\mathbb{Z}[\mu]$ are obtained.
The notion of the universal Krichever formal group over the ring $\mathcal{A}_{\mathrm{Kr}}$ is introduced; its exponential is determined by the Baker–Akhiezer function $\Phi(t)=\Phi(t;\tau,g_2,g_3)$, where $\tau$ is a point on the elliptic curve with Weierstrass parameters $(g_2, g_3)$. As a consequence, results on the Krichever genus which takes values in the ring $\mathcal{A}_{\mathrm{Kr}}\otimes \mathbb{Q}$ of polynomials in four variables are obtained. Conditions necessary and sufficient for an elliptic formal group to be a Krichever formal group are found.
A quasiperiodic function $\Psi(t)=\Psi(t; v,w, \mu)$ is introduced; its logarithmic derivative defines the exponential of the general elliptic formal group law, where $v$ and $w$ are points on the elliptic curve with parameters $\mu$. For $w\neq\pm v$, this function has the branching points $t=v$ and $t=-v$, and for $w=\pm v$, it coincides with $\Phi(t;v,g_2,g_3)$ and becomes meromorphic. An addition theorem for the function $\Psi(t)$ is obtained. According to this theorem, the function $\Psi(t)$ is the common eigenfunction of differential operators of orders 2 and 3 with doubly periodic coefficients.

Keywords: elliptic Hirzebruch genera, addition theorems, Baker–Akhiezer function, deformed Lamé equation

DOI: https://doi.org/10.4213/faa3037

Full text: PDF file (331 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2011, 45:2, 99–116

Bibliographic databases:

UDC: 517.583+517.958+512.741
Received: 29.12.2010

Citation: V. M. Buchstaber, E. Yu. Bun'kova, “Krichever Formal Groups”, Funktsional. Anal. i Prilozhen., 45:2 (2011), 23–44; Funct. Anal. Appl., 45:2 (2011), 99–116

Citation in format AMSBIB
\Bibitem{BucBun11}
\by V.~M.~Buchstaber, E.~Yu.~Bun'kova
\paper Krichever Formal Groups
\jour Funktsional. Anal. i Prilozhen.
\yr 2011
\vol 45
\issue 2
\pages 23--44
\mathnet{http://mi.mathnet.ru/faa3037}
\crossref{https://doi.org/10.4213/faa3037}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2848776}
\zmath{https://zbmath.org/?q=an:1271.55005}
\elib{http://elibrary.ru/item.asp?id=20730616}
\transl
\jour Funct. Anal. Appl.
\yr 2011
\vol 45
\issue 2
\pages 99--116
\crossref{https://doi.org/10.1007/s10688-011-0012-y}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000298226000002}
\elib{http://elibrary.ru/item.asp?id=16999888}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79958712693}


Linking options:
  • http://mi.mathnet.ru/eng/faa3037
  • https://doi.org/10.4213/faa3037
  • http://mi.mathnet.ru/eng/faa/v45/i2/p23

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. M. Bakuradze, “The formal group laws of Buchstaber, Krichever, and Nadiradze coincide”, Russian Math. Surveys, 68:3 (2013), 571–573  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. V. M. Buchstaber, S. Terzic, “Toric genera of homogeneous spaces and their fibrations”, Int. Math. Res. Not. IMRN, 2013, no. 6, 1324–1403  crossref  mathscinet  zmath  isi  elib  scopus
    5. V. M. Buchstaber, E. Yu. Netay, “$\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology”, Russian Math. Surveys, 69:4 (2014), 757–759  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. M. Bakuradze, “Computing the Krichever genus”, J. Homotopy Relat. Struct., 9:1 (2014), 85–93  crossref  mathscinet  zmath  isi  scopus
    7. M. Bakuradze, “On the Buchstaber formal group law and some related genera”, Proc. Steklov Inst. Math., 286 (2014), 1–15  mathnet  crossref  crossref  isi  elib  elib
    8. V. M. Bukhshtaber, E. Yu. Bunkova, “Universalnaya formalnaya gruppa, opredelyayuschaya ellipticheskuyu funktsiyu urovnya 3”, Chebyshevskii sb., 16:2 (2015), 66–78  mathnet  elib
    9. V. M. Buchstaber, E. Yu. Bunkova, “Manifolds of solutions for Hirzebruch functional equations”, Proc. Steklov Inst. Math., 290:1 (2015), 125–137  mathnet  crossref  crossref  isi  elib  elib
    10. V. M. Buchstaber, A. V. Ustinov, “Coefficient rings of formal group laws”, Sb. Math., 206:11 (2015), 1524–1563  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. V. M. Buchstaber, I. V. Netay, “Hirzebruch Functional Equation and Elliptic Functions of Level $d$”, Funct. Anal. Appl., 49:4 (2015), 239–252  mathnet  crossref  crossref  isi  elib
    12. E. Yu. Bunkova, V. M. Buchstaber, A. V. Ustinov, “Coefficient rings of Tate formal groups determining Krichever genera”, Proc. Steklov Inst. Math., 292 (2016), 37–62  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. E. Yu. Bunkova, “Elliptic function of level $4$”, Proc. Steklov Inst. Math., 294 (2016), 201–214  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Leclerc M.-A., “The Hyperbolic Formal Affine Demazure Algebra”, Algebr. Represent. Theory, 19:5 (2016), 1043–1057  crossref  mathscinet  zmath  isi  scopus
    15. A. V. Ustinov, “Buchstaber Formal Group and Elliptic Functions of Small Levels”, Math. Notes, 102:1 (2017), 81–91  mathnet  crossref  crossref  mathscinet  isi  elib
    16. Lenart C., Zainoulline K., “Towards Generalized Cohmology Schubert Calculus Via Formal Root Polynomials”, Math. Res. Lett., 24:3 (2017), 839–877  crossref  mathscinet  zmath  isi
    17. Lenart C., Zainoulline K., “A Schubert Basis in Equivariant Elliptic Cohomology”, N. Y. J. Math., 23 (2017), 711–737  mathscinet  zmath  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
    Number of views:
    This page:788
    Full text:216
    References:51
    First page:25

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020