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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
DOI:
https://doi.org/10.4213/faa3037
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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
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V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950
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M. Bakuradze, “The formal group laws of Buchstaber, Krichever, and Nadiradze coincide”, Russian Math. Surveys, 68:3 (2013), 571–573
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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
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V. M. Buchstaber, S. Terzic, “Toric genera of homogeneous spaces and their fibrations”, Int. Math. Res. Not. IMRN, 2013, no. 6, 1324–1403
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V. M. Buchstaber, E. Yu. Netay, “$\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology”, Russian Math. Surveys, 69:4 (2014), 757–759
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M. Bakuradze, “Computing the Krichever genus”, J. Homotopy Relat. Struct., 9:1 (2014), 85–93
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M. Bakuradze, “On the Buchstaber formal group law and some related genera”, Proc. Steklov Inst. Math., 286 (2014), 1–15
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V. M. Bukhshtaber, E. Yu. Bunkova, “Universalnaya formalnaya gruppa, opredelyayuschaya ellipticheskuyu funktsiyu urovnya 3”, Chebyshevskii sb., 16:2 (2015), 66–78
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V. M. Buchstaber, E. Yu. Bunkova, “Manifolds of solutions for Hirzebruch functional equations”, Proc. Steklov Inst. Math., 290:1 (2015), 125–137
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V. M. Buchstaber, A. V. Ustinov, “Coefficient rings of formal group laws”, Sb. Math., 206:11 (2015), 1524–1563
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V. M. Buchstaber, I. V. Netay, “Hirzebruch Functional Equation and Elliptic Functions of Level $d$”, Funct. Anal. Appl., 49:4 (2015), 239–252
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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
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E. Yu. Bunkova, “Elliptic function of level $4$”, Proc. Steklov Inst. Math., 294 (2016), 201–214
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Leclerc M.-A., “The Hyperbolic Formal Affine Demazure Algebra”, Algebr. Represent. Theory, 19:5 (2016), 1043–1057
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A. V. Ustinov, “Buchstaber Formal Group and Elliptic Functions of Small Levels”, Math. Notes, 102:1 (2017), 81–91
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Lenart C., Zainoulline K., “Towards Generalized Cohmology Schubert Calculus Via Formal Root Polynomials”, Math. Res. Lett., 24:3 (2017), 839–877
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Lenart C., Zainoulline K., “A Schubert Basis in Equivariant Elliptic Cohomology”, N. Y. J. Math., 23 (2017), 711–737
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E. Yu. Bunkova, “Universal Formal Group for Elliptic Genus of Level $N$”, Proc. Steklov Inst. Math., 305 (2019), 33–52
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Malkhaz Bakuradze, Vladimir V. Vershinin, “On Addition Theorems Related to Elliptic Integrals”, Proc. Steklov Inst. Math., 305 (2019), 22–32
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