Theta Relations Give a Plane Cubic

Let \([X_0:X_1:X_2]=[s_0(z):s_1(z):s_2(z)].\)

Here $s_0,s_1,s_2$ are degree $3$ theta sections of a very ample line bundle $L$. They define an embedding \(\phi_L:E\hookrightarrow \mathbb{P}^2.\)

Since $E$ is one-dimensional, its image is a plane curve. Since $\deg L=3$, the image has degree $3$.

Therefore the three theta sections satisfy one homogeneous cubic relation: \(F_{\tau}(X_0,X_1,X_2)=0.\)

The coefficients of $F_{\tau}$ are determined by theta constants. Concretely, products such as \(s_i(z)s_j(z)s_k(z)\) are degree $9$ theta sections, and theta addition formulas give linear relations among these products.

These relations produce the cubic equation for the image curve.

With a symmetric theta basis, the cubic can be written in Hesse form: \(X_0^3+X_1^3+X_2^3-3\lambda(\tau)X_0X_1X_2=0.\)

Here $\lambda(\tau)$ is a modular quantity determined by theta constants.