
Tr. 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 $m = [(N2)/2]$ and $n = [(N1)/2]$ there exist parameters $(a_1,…,a_m,b_1,…,b_n)$ for which the relation $\prod _{j=1}^{n1}(B(u) + b_j u)^2\cdot (B(u) + b_n u)^{N2n} = A(u)^2 \prod _{k=1}^{m1}(A(u) + a_k u^2)^2 \cdot (A(u) + a_m u^2)^{N12m}$ 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 wellknown 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.
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 Received: September 15, 2018 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, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 40–60; Proc. Steklov Inst. Math., 305 (2019), 33–52
Citation in format AMSBIB
\Bibitem{Bun19}
\by E.~Yu.~Bunkova
\paper Universal Formal Group for Elliptic Genus of Level~$N$
\inbook Algebraic topology, combinatorics, and mathematical physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 4060
\publ Steklov Math. Inst. RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3988}
\crossref{https://doi.org/10.4213/tm3988}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 305
\pages 3352
\crossref{https://doi.org/10.1134/S0081543819030039}
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