Seiberg-Witten theory is a powerful framework for understanding the exact non-perturbative dynamics of 4d $\mathcal{N} = 2$ supersymmetric QFTs. On the Coulomb branch of the moduli space, the low-energy physics is described by an abelian gauge theory with a holomorphic structure constrained by supersymmetry and duality. In this talk, I will explain the emergence of $PSL(2,\mathbb{Z})$ invariance in this effective field theory and how this naturally leads to a fibration of elliptic curves over the Coulomb branch. Focusing on the simplest case of $\mathcal{N} = 2$ SU(2) gauge theory without flavors, I will discuss the singularity structure of the Coulomb branch and the physical significance of these special points. I will conclude by briefly commenting on the central role that the singular structure of the moduli space plays in the classification of 4d $\mathcal{N}=2$ SCFTs.