  The p-adic Simpson correspondence due to Faltings is a p-adic analogue of
non-abelian Hodge theory. The following is the main result of this article: The
correspondence for line bundles can be enhanced to a rigid analytic morphism of
moduli spaces under certain smallness conditions. In the complex setting,
Simpson shows that there is a complex analytic morphism from the moduli space
for the vector bundles with integrable connection to the moduli space of
representations of a finitely generated group as algebraic varieties. We give a
p-adic analogue of Simpson's result.
