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Hida theory is a vast domain of research. I am assuming that that you are in the simplest and oldest setting: Hida theory for ordinary eigencuspforms for the group $\operatorname{GL}_2$ over $\mathbb …

In the ordinary case, the argument is simple so let me recall it here. The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^ …

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb …

It follows from deformation theory of Galois (pseudo-)representations that $\mathbb T(\Lambda)$ is a complete noetherian semilocal ring. The maximal ideals correspond to mod $p$ modular representat …

Examples of the phenomenon alluded to in Question 1 are actually plentiful. The first that came to my attention is described in M.Emerton $p$-adic families of modular forms (after Hida, Coleman, and …