The universal formal group that defines the elliptic function of level~3

V. M. Buchstaber ; E. Yu. Bunkova

Čebyševskij sbornik, Tome 16 (2015) no. 2, pp. 66-78.

Résumé

The classical theorem of M. Lazar (see [1]) on the structure of the ring of coefficients of the universal formal group is a key result of the theory of one-dimensional formal groups. The discovery of the formal group of geometric cobordisms ([2], [3]) and D. Quillen's theorem ([4]) that it can be identified with the universal formal group allowed to introduce the theory of formal groups in the apparatus of algebraic topology, including the apparatus of the theory of Hirzebruch genera. Due to this there has been a widely-known fundamental mutual penetration of methods and results of algebraic topology, (see [5]), algebraic geometry, the theory of functional equations and mathematical physics. Important applications in algebraic topology found results of the theory of elliptic functions and Baker–Akhiezer functions, which play a fundamental role in the modern theory of integrable systems. The construction of universal formal groups of given form, with exponents given by these functions, became actual. Known results in this direction use both classic and recently obtained addition theorems, that determine the form of formal groups. In this paper we solved a long standing problem: we have found the form of universal formal group the exponent of which is the elliptic function of level 3. We have obtained results on the coefficient ring of this group and described its relationship with known universal formal groups. Bibliography: 15 titles.

Détail

Keywords: formal groups, elliptic function of level 3.

Bibliographie

[1] M. Lazard, “Sur les groupes de Lie formels a un parametre”, Bull. Soc. Math. France, 83 (1955), 251–274 | MR | Zbl

[2] Buhstaber V. M., Miscenko A. S., Novikov S. P., “Formal groups and their role in the apparatus of algebraic topology”, Uspehi Mat. Nauk, 26:2 (1971), 131–154 (Russian) | MR | Zbl

[3] Novikov S. P., “Methods of algebraic topology from the point of view of cobordism theory”, Izv. Akad. Nauk SSSR Ser. Mat., 31:4 (1967), 855–951 (Russian) | MR | Zbl

[4] D. Quillen, “On the formal group laws of unoriented and complex cobordism theory”, Bull. Amer. Math. Soc., 75:6 (1969), 1293–1298 | DOI | MR | Zbl

[5] Bukhshtaber V. M., “Complex cobordisms and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950 | DOI | DOI | MR | Zbl

[6] E. T. Whittaker, G. N. Watson, A Course in Modern Analysis, 4th ed., Cambridge University Press, Cambridge, England, 1990

[7] Buchstaber V. M., Ustinov A. V., “Rings coefficients formal groups”, Mat. Sb.

[8] Bukhshtaber V. M., “Functional equations that are associated with addition theorems for elliptic functions, and two-valued algebraic groups”, Russian Math. Surveys, 45:3 (1990), 213–215 | DOI | MR | Zbl

[9] F. Hirzebruch, Elliptic genera of level $N$ for complex manifolds, Prep. MPI, 88–24

[10] F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular Forms, Vieweg, Braunschweig, 1992 | MR | Zbl

[11] I. M. Krichever, “Generalized elliptic genera and Baker–Akhiezer functions”, Math. Notes, 47:2 (1990), 132–142 | DOI | MR | Zbl

[12] J. Barr von Oehsen, “Elliptic genera of level $N$ and Jacobi polynomials”, Proc. Amer. Math. Soc., 122 (1994), 303–312 | MR | Zbl

[13] M. Hazewinkel, Formal Groups and Applications, Academic Press, New York–San Francisco–London, 1978 | MR | Zbl

[14] Bukhshtaber V. M., Bun'kova E. Yu., “Krichever formal groups”, Funct. Anal. Appl., 45:2 (2011), 99–116 | DOI | DOI | MR

[15] Buchstaber V. M., Netay E. Yu., “$\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology”, Russian Math. Surveys, 69:4 (2014), 757–759 | DOI | DOI | MR | MR | Zbl