What is the difference between Hida and Coleman families?

Question

From my understanding: Hida families and Coleman families of modular forms are roughly given by $p$-adic modular forms whose $q$-expansion at classical weights is "close" to a $q$-expansion of a classical modular form. But the difference between the two families is not exactly clear to me; my understanding is only that there is a difference in the $p$-adic valuation of the $a_n$ coefficients...

An additional question: is the theory of Coleman families for general automorphic forms developed? For example for Hilbert modular forms would already be useful.

Answer 1

The difference is the generality of the setting: Hida families (first introduced by Hida in the early 80s) apply only to eigencuspforms which are so-called ordinary at $p$ (roughly speaking, the $p$-adic valuation of $a_p$ is zero), Coleman families (constructed about 10 years later by Coleman) apply to eigencuspforms with so-called finite slope (roughly speaking the $p$-adic valuation of $a_p$ is bounded, that is to say $a_p≠0$). In particular, the Hida family passing through an ordinary classical eigencuspform is a Coleman family but the reverse need not necessarily be true (in fact, even the direct statement might not be strictly true, depending on your precise definitions of these families).

The theory of Coleman families has certainly been extended to many settings beyond classical modular forms, in particular to Hilbert modular forms (by constructions of Buzzard, Kisin-Lai, Kassaei, Emerton, Urban, Brasca, Andreatta-Iovita-Pilloni etc.) and to automorphic representations of other reductive groups. Each of these extensions requires a lot more explanation.