On $p$-adic intermediate Jacobians

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 $\bf {C}$. 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.