Cubic Equation from Theta Functions

Start with the degree $3$ theta basis: \(s_k(z)= \sum_{n\in\mathbb Z} \exp\left( 3\pi i(n+k/3)^2\tau +6\pi i(n+k/3)z \right), \quad k=0,1,2.\)

These give projective coordinates: \([X_0:X_1:X_2] = [s_0(z):s_1(z):s_2(z)].\)

Now look at cubic expressions in $X_0,X_1,X_2$: \(X_0^3,\ X_0^2X_1,\ \ldots,\ X_2^3.\)

There are $10$ such monomials. But each one is a degree $9$ theta section, and \(\dim H^0(E,L^{\otimes 3})=9.\)

So the $10$ cubic expressions must satisfy one linear relation: \(\sum_{i+j+k=3} a_{ijk}(\tau)X_0^iX_1^jX_2^k=0.\)

A symmetric theta basis means that the $3$-torsion symmetries of the torus act on the coordinates in the standard way: \((X_0,X_1,X_2)\mapsto (X_1,X_2,X_0),\) and \((X_0,X_1,X_2)\mapsto (X_0,\zeta X_1,\zeta^2X_2), \qquad \zeta^3=1.\)

The cubic relation must be invariant under these symmetries. The diagonal symmetry leaves only \(X_0^3,\quad X_1^3,\quad X_2^3,\quad X_0X_1X_2.\)

The cyclic symmetry forces the coefficients of $X_0^3,X_1^3,X_2^3$ to be equal. Thus the relation has the form \(A(\tau)(X_0^3+X_1^3+X_2^3)+B(\tau)X_0X_1X_2=0.\)

After rescaling, this becomes the Hesse cubic: \(X_0^3+X_1^3+X_2^3 -3\lambda(\tau)X_0X_1X_2=0.\)

The coefficient is determined by theta constants. For example, evaluating at $z=0$ gives \(\lambda(\tau) = \frac{s_0(0)^3+s_1(0)^3+s_2(0)^3} {3s_0(0)s_1(0)s_2(0)}.\)