We will begin with an overview of Stark's conjectures before discussing the case of imaginary quadratic fields, covering both the limit formula and the existence of elliptic units. The classical expositions of these are at times lacking in intuition, but thanks to Kato's deep insights 20 years ago, we can present more geometric and illuminating proofs of both results.

Fix an Artin representation rho. Work in progress by Emmanuel Lecouturier and Loïc Merel claims that the special values L(f,rho,1) for certain modular forms f see some global data related to the L-function attached to rho. We first give a brief exposition on Mazur’s Eisenstein ideal, which lies at the heart of their work. We then describe this conjectural phenomenon in a few simple cases, the last being related to a conjecture of Harris and Venkatesh.