  Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete
discrete valuation ring with residue field $k$ and field of fractions $K$ of
characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is
smooth over $k$, we partially prove here a conjecture of Berthelot about the
overconvergence of the higher direct images of the structure sheaf under a
proper smooth morphism $f:X\to S$; when $k$ is perfect and $\mathcal{V}$ is
tamely ramified such direct images are always convergent, not only for the
structure sheaf but also for (almost) every convergent $F$-isocrystals. More
generally, we prove this overconvergence when $f$ is liftable over
$\mathcal{V}$, or when $X$ is a relative complete intersection in some
projective spaces over $S$, and taking as coefficients any overconvergent
isocrystals. We then apply these results to $L$-functions with coefficients
such direct images with Frobenius structure: we derive rationality or
meromorphy for these $L$-functions (Dwork's conjecture), and we study their
$p$-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns
the ordinary abelian schemes. A more precise presentation of results by
chapters is given in the introduction.
