  In this paper we investigate locally free representations of a quiver Q over
a commutative Frobenius algebra R by arithmetic Fourier transform. When the
base field is finite we prove that the number of isomorphism classes of
absolutely indecomposable locally free representations of fixed rank is
independent of the orientation of Q. We also prove that the number of
isomorphism classes of locally free absolutely indecomposable representations
of the preprojective algebra of Q over R equals the number of isomorphism
classes of locally free absolutely indecomposable representations of Q over
R[t]/(t^2). Using these results together with results of Geiss, Leclerc and
Schroer we give, when k is algebraically closed, a classification of pairs
(Q,R) such that the set of isomorphism classes of indecomposable locally free
representations of Q over R is finite. Finally, when the representation is free
of rank 1 at each vertex of Q, we study the function that counts the number of
isomorphism classes of absolutely indecomposable locally free representations
of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial
in q and their generating function is rational and satisfies a functional
equation.
