  In this thesis, we study counts of quiver representations over finite rings
of truncated power series. We prove a plethystic formula relating counts of
quiver representations over these rings and counts of jets on fibres of quiver
moment maps. This solves a conjecture of Wyss and allows us to compute both
counts on additional examples, using local zeta functions. The relation between
counts of representations and counts of jets generalises the relation between
Kac polynomials and counts of points on preprojective stacks. Pursuing this
analogy, we establish further properties of our counts. We show that, for
totally negative quivers, counts of jets converge to p-adic integrals on fibres
of quiver moment maps. One expects a relation between these p-adic integrals
and BPS invariants of preprojective algebras i.e. Kac polynomials. For small
rank vectors, we also prove that the polynomials counting indecomposable quiver
representations over finite rings have non-negative coefficients. Moreover, we
show that jet schemes of fibres of quiver moment maps are cohomologically pure
in that setting, so that their Poincar\'e polynomials are given by the former
counts. This is reminiscent of the structure of cohomological Hall algebras,
which are built from the cohomology of preprojective stacks. Finally, we
compute the cohomology of jet spaces of preprojective stacks explicitly for the
A2 quiver. Building on the structure of the preprojective cohomological Hall
algebra of A2, we propose a candidate analogue of the BPS Lie algebra and
conjecture the existence of a Hall product on the cohomology of these jet
spaces.
