Height Pairings

generalizing the Neron-Tate pairing on abelian varieties. Note that our cycles are of a dimension where their expected intersection has dimension −1. Example 1.1 ([9], [3]) Let C/K be a smooth projective curve, with ∞ ∈ C(K) giving i : C ↪→ Pic(C). Let 〈·, ·〉NT : Pic(C)(K)×Pic(C)(K)→ R be the Neron-Tate height pairing, identifying P̂ic(A) ∼= Pic(A) via the theta divisor. Then, once we have defined 〈·, ·〉, we will have 〈P −∞, Q−∞〉 = 〈i(P ), i(Q)〉NT . Example 1.2 More generally, for an abelian variety A/K of dimension n, the maps CH(A)0 ∼= Â(K), CH(A)0 → A(K) identify the pairings 〈·, ·〉 with 〈·, ·〉NT : A(K)× Â(K)→ R. Recall that Neron-Tate pairings came from the canonical height function associated to the Poincare bundle. The Poincare bundle is not ample, although it is when restricted to the “diagonal” of A× Â via a polarization A → Â. In particular, we get an induced pairing A(K)× A(K) → R which is symmetric and non-degenerate modulo torsion.