We present some recent results on the first eigenvalue of the magnetic Laplacian associated with closed potential 1-forms on compact Riemannian surfaces. We introduce some (conformal) spectral invariants, naturally associated with the first eigenvalue, which can be estimated by the geometry of the Jacobian torus of the surface. In genus 1 the computations are explicit, and one can have a simple geometric interpretation of the invariants. Finally, we present our main result: we define the notion of ground state spectrum and a corresponding notion of "ground state isospectrality", and we prove that if two Riemannian metrics are ground state isospectral, then they are conformal and have the same volume. Joint work with Bruno Colbois (Université de Neuchâtel) and Alessandro Savo (Sapienza Università di Roma).