elliptic generalizations of Euler's trick

Notice that, as an alternative to using the product formula for the sine function to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$, one can just as easily work with the partial fraction decomposition of the cotangent function (obtained by logarithmically differentiating said product formula). Indeed, by differentiating everything one more time, we can evaluate $\zeta(2n)$, for $n\in\mathbb{N},$ by comparing the partial fraction decomposition of the reciprocal of $\sin^2 z$ with its Laurent expansion.

Now the elliptic analog of the reciprocal of $\sin^2 z$ is Weierstrass's elliptic function (Weierstrass's $\wp$-function). So one needs to compare the partial fraction decomposition and the Laurent expansion of the $\wp$-function - I recommend the MathWorld article for details on this. There it shows that the coefficients in the Laurent expansion of the $\wp$-function may be expressed as a polynomial with rational coefficients in terms of 2 quantities known as the elliptic invariants - essentially the elliptic analogs of $\zeta(4)$ and $\zeta(6)$ (see equations (24) and (25)). Thus the elliptic analog of $\zeta(2n)$ is a polynomial (with rational coefficients) in the two elliptic invariants.

This is then an elliptic analog of Euler's result that $\zeta(2n)$ is a rational multiple of the n-th power of $\zeta(2).$