Elliptic function of level 4

Abstract

The work is dedicated to the theory of elliptic functions of level $n$. An elliptic function of level $n$ determines a Hirzebruch genus that is called elliptic genus of level $n$. Elliptic functions of level $n$ are also interesting as solutions of Hirzebruch functional equations. The elliptic function of level $2$ is the Jacobi elliptic sine. It determines the famous Ochanine--Witten genus. It is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 -v^2}{u B(v) - v B(u)}, \quad B(0) = 1. \] The elliptic function of level $3$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u A(v)^2 - v A(u)^2}, \qquad A(0) = 1, \quad A"(0) = 0. \] In this work we have obtained that the elliptic function of level $4$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u B(v)-v B(u)}, \text{ where } A(0) = B(0) = 1, \] and for $B^{\prime }(0)=A"(0)=0,A^{\prime }(0)=A_1,B"(0)=2B_2$ the relation holds \[ (2 B(u) + 3 A_1 u)^2 = 4 A(u)^3 - (3 A_1^2 - 8 B_2) u^2 A(u)^2. \] To prove this result we have expressed the elliptic function of level $4$ in terms of Weierstrass elliptic functions.