Elliptic function of level $4$

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.