Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions

We construct an Euler system of $p$-adic zeta elements over the eigencurve which interpolates Kato's zeta elements over all classical points. Applying a big regulator map gives rise to a purely algebraic construction of a two-variable $p$-adic $L$-function over the eigencurve. As a first application of these ideas, we prove the equality of the $p$-adic $L$-functions associated with a critical-slope refinement of a modular form by the works of Bella\"iche/Pollack-Stevens and Kato/Perrin-Riou.