  For an algebraic variety $X$ of dimension $d$ with totally degenerate
reduction over a $p$-adic field (definition recalled below) and an integer $i$
with $1\leq i\leq d$, we define a rigid analytic torus $J^i(X)$ together with
an Abel-Jacobi mapping to it from the Chow group $CH^i(X)_{hom}$ of codimension
$i$ algebraic cycles that are homologically equivalent to zero modulo rational
equivalence. These tori are analogous to those defined by Griffiths using Hodge
theory over the complex numbers. We compare and contrast the complex and
$p$-adic theories. Finally, we examine a special case of a $p$-adic analogue of
the Generalized Hodge Conjecture
