There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}$$
A beautiful solution [1] to this is given by the third Jacobi theta function $\theta_{3}$:
$$K(k) = \frac{\pi}{2} \theta_{3}^{2}(q)$$
for some function $q = q(k)$.
This means that $K(k)$ enjoys some sort of modularity after some clever transformation! And whenever I see a function with (near) modularity, I wonder if it's a function (or a section) on the moduli space of elliptic curves.
Question: Given $k$, can you construct a Riemann surface $\Sigma(k)$ and interpret $K(k)$ as some quantity related to $\Sigma(k)$ (e.g. a period or some sort) so that the modularity of $K(k)$ is a direct corollary? It would be even better if such $\Sigma(k)$ is the surface defined by
$$y^{2} = (1-x^{2})(1-k^{2}x^{2}).$$
