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 Tr. Mat. Inst. Steklova, 2016, Volume 294, Pages 216–229 (Mi tm3728)

Elliptic function of level $4$

E. Yu. Bunkova

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

Abstract: The article is devoted to the theory of elliptic functions of level $n$. An elliptic function of level $n$ determines a Hirzebruch genus called an elliptic genus of level $n$. Elliptic functions of level $n$ are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level $2$ is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form $F(u,v)=(u^2-v^2)/(uB(v)-vB(u))$, $B(0)=1$. The elliptic function of level $3$ is the exponential of the universal formal group of the form $F(u,v)=(u^2A(v)-v^2 A(u))/(uA(v)^2-vA(u)^2)$, $A(0)=1$, $A"(0)=0$. In the present study we show that the elliptic function of level $4$ is the exponential of the universal formal group of the form $F(u,v)=(u^2A(v)-v^2A(u))/(uB(v)-vB(u))$, where $A(0)=B(0)=1$ and for $B'(0)=A"(0)=0$, $A'(0)=A_1$, and $B"(0)=2B_2$ the following relation holds: $(2B(u)+3A_1u)^2=4A(u)^3-(3A_1^2-8B_2)u^2A(u)^2$. To prove this result, we express the elliptic function of level $4$ in terms of the Weierstrass elliptic functions.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968516030122

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 294, 201–214

Bibliographic databases:

Document Type: Article
UDC: 512.741+515.178.2+517.965

Citation: E. Yu. Bunkova, “Elliptic function of level $4$”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Tr. Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 216–229; Proc. Steklov Inst. Math., 294 (2016), 201–214

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. V. Ustinov, “Buchstaber Formal Group and Elliptic Functions of Small Levels”, Math. Notes, 102:1 (2017), 81–91
2. Elena Yu. Bunkova, “Hirzebruch functional equation: classification of solutions”, Proc. Steklov Inst. Math., 302 (2018), 33–47
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