Title: Dirichlet L-series at s = 0 and the scarcity of Euler systems

Abstract: Ever since their introduction, Euler systems have played an important role in understanding the mysterious link between analysis and arithmetic that manifests itself in leading term formulae for L-series. In this talk, I will discuss joint work with Burns, Daoud, and Seo that proves a precise description (conjectured by Coleman) of the set of all Euler systems over the rationals. This also leads to a new approach towards leading term conjectures, including a proof of the `minus part’ of the equivariant Tamagawa Number Conjecture outside 2.