Let $M = (\mathbb{Z}\_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL\_2(\mathbb{Z}\_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(G, M)$ (or more interesting to me, $H^1\_f(G, M)$) is to perturbations in $\rho$.


For example, say I replace $\rho$ by $\rho'$, with $\overline{\rho} = \overline{\rho'}$, i.e. the residual representations are isomorphic.


Denote by $M'$ the Galois module $(\mathbb{Z}\_p)^2$ with the action $\rho'$. Is it true that
$H^1(G, M)\cong H^1(G, M')$?


Thanks in advance!



Deformations of Galois cohomology
The answer to your example question is "No".


Let $G$ be isomorphic to $\widehat{\mathbf{Z}}$, and $\phi$ the topological generator of $G$ (corresponding to $1 \in \widehat{\mathbf{Z}}$). Then one computes $H^0(G, M) = M^{\phi = 1}$ and $H^1(G, M) = M / (\phi - 1)M$, where $\phi$ is the generator of $G$.


Now $M \cong \mathbf{Z}\_p$ be the trivial one-dimensional representation, and let $M'$ be the unramified representation with Frobenius acting as $1 + p$. Then $H^1(G, M) = M = \mathbf{Z}\_p$, whereas $H^1(G, M') = \mathbf{F}\_p$.


(In this case, we do at least have $H^1(G, M) \otimes \mathbf{F}\_p = H^1(G, M') \otimes \mathbf{F}\_p$. But even this is just a small-number coincidence, coming from the fact that the $H^2$'s vanish, and will fail if you take more complicated groups.)


What *is* true is that if $M \cong M' \bmod p$ (and $G$ has reasonable finiteness properties), then the cohomology *complexes* $R\Gamma(G, M)$ and $R\Gamma(G, M')$ are "congruent mod $p$ in the derived category", in the sense that $R\Gamma(G, M) \otimes^{\mathbf{L}} \mathbf{F}\_p = R\Gamma(G, M') \otimes^{\mathbf{L}} \mathbf{F}\_p = R\Gamma(G, \overline{M})$ (where $\overline{M}$ is the common residual representation). But much can go wrong in translating that into a concrete statement about cohomology groups.


The situation for $H^1\_f$ is worse – far, far worse – because in general there is no sensible intrinsic definition of $H^1\_f$ of a mod $p$ representation. (A large part of the machinery of Iwasawa theory is designed to measure, control, and in some cases cleverly exploit, the discrepancy between $H^1\_f$'s of congruent representations).


(EDIT: You might find the [Greenberg-Vatsal paper](https://arxiv.org/abs/math/9906215) interesting in this line -- it shows a result which roughly amounts to "Selmer groups of congruent modular forms are congruent", but there is a *huge* amount of work involved in making this precise and avoiding all the technical pitfalls.)


