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Funktsional. Anal. i Prilozhen.:

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

This article is cited in 18 scientific papers (total in 19 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


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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
\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
\jour Funct. Anal. Appl.
\yr 2011
\vol 45
\issue 2
\pages 99--116

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    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
    18. E. Yu. Bunkova, “Universal Formal Group for Elliptic Genus of Level $N$”, Proc. Steklov Inst. Math., 305 (2019), 33–52  mathnet  crossref  crossref  mathscinet  isi  elib
    19. Malkhaz Bakuradze, Vladimir V. Vershinin, “On Addition Theorems Related to Elliptic Integrals”, Proc. Steklov Inst. Math., 305 (2019), 22–32  mathnet  crossref  crossref  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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