  Let $k/\mathbb F_p$ denote a finite field. For any split connected reductive
group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois
representations $\overline\rho:Gal(\overline F/F) \to G(k)$ to continuous
families $X_{\overline\rho}$ of Galois representations $Gal(\overline F/F) \to
G(\overline{\mathbb Q_p})$ lifting $\overline\rho$ such that the space of
points of $X_{\overline\rho}$ which are geometric (in the sense of the
Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive
codimension in $X_{\overline\rho}$. Thus the set of points in
$X_{\overline\rho}$ which could (conjecturally) be associated to automorphic
forms is sparse. This generalizes a result of Calegari and Mazur for $F/\mathbb
Q$ quadratic imaginary and $G = GL_2$. The sparsity of automorphic points for
$F$ a CM field contrasts with the situation when $F$ is a totally real field,
where automorphic points are often provably dense.
