Modular Jacobians and degenerate p-adic geometry

p-Adic analogues of triangulations of Riemann surfaces give us a very concrete way of understanding degenerate parts of modular Jacobians. In this talk, I will discuss how this yields a flexible way to understand the action of Hecke operators on modular curves, and functoriality of canonical integral "hidden" structures on de Rham cohomology. Finally, I will discuss progress on a strategy for defining p-adic L-functions of special modular forms via such degenerate techniques, proposed by Schneider.