I am trying to learn about deformations of degree 2 Galois representations mod $p$ and get a grasp of simple intuitions on examples.
In basic references the explicit examples of universal deformation rings have Krull dimension 4 and thus the universal deformation space has dimension 4 as an algebraic variety -relative dimension 3 over Spec $\mathbb Z_p$. But then one treats various restrictions on the deformations, comprising a "deformation type". For instance one fixes the determinant of deformations, one requires a certain upper triangular form ("minimal ramification" in the terminology of Mazur), and one requires no ramification outside a set of primes $\Sigma$ denote the resulting type $(\mathcal D,\Sigma)$. As I understand the first 2 restrictions cut down the "degrees of freedom" by 3 so that the universal deformation rings of type $\mathcal D$, $R_\mathcal D$ are finite $\mathbb Z_p$-modules and we have only a finite number of possible deformations -corresponding to finitely many newforms yielding the original mod $p$ Galois representation. Another restriction is asking the liftings to be unramified outside some set $\Sigma$. So as we increase $\Sigma$ we increase possible ramification, possible deformations. The unrestricted case corresponds to taking $\Sigma$ the set of all primes.
I have several questions: can you confirm restricting the determinant cuts down the dimension of the universal deformation space by 1, and asking minimal ramification by 2?
Can you tell me what kind of new representations adding possible ramification primes, increasing $\Sigma$, adds to the set of all liftings of type $(\mathcal D,\Sigma)$, the image of the deformation functor on $R_{\mathcal D,\Sigma}$? Does it affect the $\mathbb Z_p$-rank of $R_{\mathcal D,\Sigma}$ or does it only affect the "obstruction equations"? Can one obtain the same universal deformation ring in 2 different ways? If ramification restriction affects the rank can a restriction of the determinant of liftings be replaced by an appropriate ramification restriction?
Also, is there some impossible additional ramification for liftings of a mod $p$ representation with given ramification? Or can one construct liftings with arbitrary additional ramification to that mod $p$?
Thank you for any indication or reference.