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 Trudy Mat. Inst. Steklova, 2019, Volume 305, Pages 40–60 (Mi tm3988)

Universal Formal Group for Elliptic Genus of Level $N$

E. Yu. Bunkova

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: An elliptic function of level $N$ determines an elliptic genus of level $N$ as a Hirzebruch genus. It is known that any elliptic function of level $N$ is a specialization of the Krichever function that determines the Krichever genus. The Krichever function is the exponential of the universal Buchstaber formal group. In this work we give a specialization of the Buchstaber formal group such that this specialization determines formal groups corresponding to elliptic genera of level $N$. Namely, an elliptic function of level $N$ is the exponential of a formal group of the form $F(u,v) =(u^2 A(v) - v^2 A(u))/(u B(v) - v B(u))$, where $A(u),B(u)\in \mathbb C[[u]]$ are power series with complex coefficients such that $A(0)=B(0)=1$, $A"(0)=B'(0)=0$, and for $m = [(N-2)/2]$ and $n = [(N-1)/2]$ there exist parameters $(a_1,…,a_m,b_1,…,b_n)$ for which the relation $\prod _{j=1}^{n-1}(B(u) + b_j u)^2\cdot (B(u) + b_n u)^{N-2n} = A(u)^2 \prod _{k=1}^{m-1}(A(u) + a_k u^2)^2 \cdot (A(u) + a_m u^2)^{N-1-2m}$ holds. For the universal formal group of this form, the exponential is an elliptic function of level at most $N$. This statement is a generalization to the case $N>2$ of the well-known result that the elliptic function of level $2$ determining the elliptic Ochanine–Witten genus is the exponential of a universal formal group of the form $F(u,v) =(u^2 - v^2)/(u B(v) - v B(u))$, where $B(u)\in \mathbb C[[u]]$, $B(0)=1$, and $B'(0)=0$. We prove this statement for $N=3,4,5,6$. We also prove that the elliptic function of level $7$ is the exponential of a formal group of this form. Universal formal groups that correspond to elliptic genera of levels $N=5,6,7$ are obtained in this work for the first time.

 Funding Agency Grant Number Contest «Young Russian Mathematics» The work was supported in part by the Young Russian Mathematics award.

DOI: https://doi.org/10.4213/tm3988

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 33–52

Bibliographic databases:

UDC: 512.741+515.178.2+517.583
Revised: December 6, 2018
Accepted: December 20, 2018

Citation: E. Yu. Bunkova, “Universal Formal Group for Elliptic Genus of Level $N$”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 40–60; Proc. Steklov Inst. Math., 305 (2019), 33–52

Citation in format AMSBIB
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