Arithmetic points are dense on a Hida family

$$\DeclareMathOperator\Spec{Spec}$$I am reading the paper "Constancy of the adjoint L-invariant" by H. Hida (J. Number Th., 2011, DOI link).

Correct me if I'm wrong, but I've read/heard that the arithmetic points $$p \in \Spec(\mathbb{I})$$ are Zariski dense (Where $$\Spec(\mathbb{I})$$ is a connected component of $$\Spec(\mathbf{h})$$, the spectrum of the universal $$p$$-ordinary Hecke Algebra). Once we have defined the L-invariant on the dense points, there is some interpolation trick to define it on all points.

How can I understand these two facts?

Also, reference request:

I've been reading through Hida's papers on $$\mathbf{h}$$ between '81-'86, as well as his course notes, textbooks, etc. This paper I linked at the top (constancy of L-invariant) is one of the most understandable expositions I have read (Due to the algebraic geometry style). Are there any modern expositions of Hida theory floating around somewhere that one can learn from? I am not familiar with motives, a language I have seen appear in Hida's work/books, so something that avoids extra-technical definitions would be great.

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 $\mathbb Q$ (the arguments I am giving would apply with little modification to $\operatorname{GL}_2$ over a totally real field $F$ after an application of the Jacquet-Langlands correspondence).

In that setting the Hida-Hecke algebra $\mathbf T$ injects in the endomorphism ring of a module which is free of finite rank over $\Lambda$ (the usual Hida algebra of diamond operators), namely the ordinary part of the inverse limit on the level at $p$ of the first cohomology group of the modular curves. So $\mathbf T$ is finite, torsion-free as $\Lambda$-module and so the ring-extension $\mathbf T/P$ of $\Lambda$ for $P$ a minimal prime of $\mathbf T$ is integral.

Hence the Zariski-density of arithmetic primes in $\mathbf T/P$ reduces to the density of primes below them in $\Lambda$ which are, by definition, the principal ideals $(\gamma-\chi(\gamma)\gamma^{k-2})$ for $\chi$ a finite order character and $\gamma$ a topological generator of $\Gamma=1+p\mathbb Z_p$. These are dense in $\Lambda=\mathbb Z_{p}[[\Gamma]]$.

These arguments carry over for many reductive groups (symplectic groups, unitary groups, units in quaternion algebras...) over totally real fields but it is typically not the case that arithmetic points are dense in the spectrum of the Hida-Hecke algebra for a reductive group over a number field which is not totally real. This is already false, for instance, for $\operatorname{GL}_2$ over a quadratic imaginary field $K$.

Regarding your question on the interpolation of the $\mathcal L$ invariant of the adjoint representation, this is a very subtle result, especially in the generality you consider here. It requires first deep modularity results of Wiles, Taylor-Wiles, Skinner-Wiles and Kisin on deformations of Galois representations with coefficients in a finite extension of $\mathbb Q_p$ (so with infinite residue field). These results establish the vanishing of some Selmer group for the adjoint representation and this in turns lead to an explicit computation of the $\mathcal L$ invariant which shows that it extends to an analytic function on the set of height-one primes of the universal Hecke algebra.

Finally, regarding your reference request, I like Control of nearly ordinary Hecke algebras (H.Hida) and Residually reducible representations and modular forms (C.Skinner, A.Wiles) Inst. Hautes Études Sci. Publ. Math. No. 89 (1999), 5--126 (2000) myself, but there are dozens of excellent references (none of which make essential use of the notion of motives, so don't get scared by that word and just mentally translate it to geometric Galois representations or compatible systems of geometric Galois representations whenever it appears).

Very late response, but I wanted to add some references here that I found helpful when learning Hida theory. Some of these are not very easily searchable on Google so I stumbled upon them by accident while looking around.

1. Eknath Ghate has a wonderful 4-part lecture series about Hida theory with lots of numerical examples. Here is the link to the first video: https://www.youtube.com/watch?v=xUNjj8-yca8

2. These notes by Kumar, Bhanarjee, and Ghate complement the above lecture videos very well. They also explain in broad strokes how Hida theory was used to prove the Iwasawa main conjecture.

3. The master's thesis "Hida families of $$p$$-adic modular forms" by Luca Marannino (searchable on google) explains Hida theory from the ground up, with almost complete proofs. It was also quite readable :)