  Given a complex quasi-projective normal variety $X$ and a linear
representation $\varrho:\pi_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of
positive characteristic, we mainly establish the following results:
  1. the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm
Sh}_\varrho(X)$ associated with $\varrho$.
  2. In cases where $X$ is projective, $\varrho$ is faithful and the
$\Gamma$-dimension of $X$ is at most two (e.g. $\dim X=2$), we prove that the
Shafarevich conjecture holds for $X$.
  3. In cases where $\varrho$ is big, we prove that the Green-Griffiths-Lang
conjecture holds for $X$.
  4. When $\varrho$ is big and the Zariski closure of $\varrho(\pi_1(X))$ is a
semisimple algebraic group, we prove that $X$ is pseudo Picard hyperbolic, and
strongly of log general type.
  5. If $X$ is special or $h$-special, then $\varrho(\pi_1(X))$ is virtually
abelian.
  We also prove Claudon-H\"oring-Koll\'ar's conjecture for complex projective
manifolds with linear fundamental groups of any characteristic.
