On 𝑝-adic intermediate Jacobians

For an algebraic variety X X of dimension d d with totally degenerate reduction over a p p -adic field (definition recalled below) and an integer i i with 1 ≤ i ≤ d 1\leq i\leq d , we define a rigid analytic torus J i ( X ) J^i(X) together with an Abel-Jacobi mapping to it from the Chow group C H i ( X ) h o m CH^i(X)_{hom} of codimension i 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 C \bf {C} . We compare and contrast the complex and p p -adic theories. Finally, we examine a special case of a p p -adic analogue of the Generalized Hodge Conjecture.