A geometric view on Iwasawa theory

These notes expand on the presentation given by the second author at the Iwasawa 2019 conference in Bordeaux of our joint work on the geometry of the $p$-adic eigencurve at a weight one CM form $f$ irregular at $p$, namely its implications in Iwasawa and in Hida theories. Novel features include the determination of -- the Fourier coefficients of the infinitesimal deformations of $f$ along each Hida family containing it in terms of $p$-adic logarithms of algebraic numbers. -- the "mysterious" cross-ratios of the ordinary filtrations of the Hida families containing $f$.