Let $K$ be a local field and $A$ an abelian variety over $K$. We take it to be minimal in the sense of Néron. In the paper Abelian varieties over large algebraic fields by Frey and Jarden, there is the following passage:
Let $p_K$ be the maximal ideal of the ring of integers of $K$ with respect to the valuation $v$. For every integer $\mu\geq 0$, denote by $\rho^{\mu}$ the map induced by the reduction modulo $p_K^{\mu + 1}$ (we use here the notations of Néron). Then $\rho^{\mu}(A)$ is a commutative algebraic group defined over $k$, and there exist natural epimorphisms $\theta^{\mu}: \rho^{\mu+1}(A(K))\rightarrow \rho^{\mu}(A(K))$, defined over the residue field $k$, whose kernel is canonically isomorphic to $(k^+)^r$ (where $k^+$ is the additive group of $k$) such that $A(K)$ is the projective limit of the sequence $$\rho^0(A(K))\xleftarrow{\theta^0} \rho^1(A(K))\xleftarrow{\theta^1} \dots \leftarrow \rho^{\mu}(A(K)) \xleftarrow{\theta^{\mu}} \rho^{\mu+1}(A(K))\leftarrow \dots $$
I don't understand the maps $\rho^{\mu}$ for $\mu>0$. Certainly there is a reduction mod $p$ map for an abelian variety defined over a DVR, but how would you reduce mod $p^2$. The obvious thing to do would be to reduce the coefficients of $A$ mod $p^2$, but why would this resulting object be defined over $k$?
I've found the relevant pages in Néron's original article, but my French isn't good enough to entirely make out what's going on.