On p-adic height pairings

Abstract

The aim of the present work is to construct p-adic height pairings in a sufficiently general setting, namely for Selmer groups of reasonably behaved p-adic Galois representations over number fields. These pairings should, modulo general conjectures on etale cohomology, induce height pairings between (homologically trivial) algebraic cycles on every proper smooth variety defined over a number field. For smooth projective varieties with good reduction at all primes dividing p, our construction requires only one cohomological assumption: we have to assume that the purity conjecture for certain etale cohomology group holds at all places of bad reduction of the given variety. More specifically, let X be a proper smooth variety, of dimension d, defined over a number field K. Write CH(X)0 for the group of homologically trivial algebraic cycles of codimension i on X, defined over K, modulo rational equivalence. If i+ j = d+ 1, then there are “etale Abel-Jacobi maps”