  A major theme in the theory of $p$-adic deformations of automorphic forms is
how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these
eigenvarieties. In this article we prove results in this vein for the ordinary
part of the eigencurve (i.e. Hida families). We address how Taylor expansions
of one variable $p$-adic $L$-functions varying over families can detect "bad"
geometric phenomena: crossing components of a certain intersection multiplicity
and ramification over the weight space. Our methods involve proving a converse
to a result of Vatsal relating congruences between eigenforms to their
algebraic special $L$-values and then $p$-adically interpolating congruences
using formal models.
