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On $$p$$-adic height pairings. (English) Zbl 0859.11038
David, Sinnou (ed.), Séminaire de théorie des nombres, Paris, France, 1990-1991. Basel: Birkhäuser. Prog. Math. 108, 127-202 (1993).
This paper extends the theory of $$p$$-adic heights to a quite general motivic setting. Starting with a suitable $$p$$-adic Galois representation $$V$$ over a number field $$K$$, Bloch and Kato have defined an analogue, $$H^1_f (K,V)$$, of the classical Selmer group for an elliptic curve. This paper defines a height pairing $H^1_f(K,V) \times H^1_f \bigl(K,V^*(1) \bigr) \to \mathbb{Q}_p.$ The construction depends on choices of a $$p$$-adic logarithm over $$K$$ and of splittings of certain Hodge filtrations associated to $$V$$.
After constructing this height pairing, the paper shows that the given height can be expressed as a sum of local heights, investigates its behavior with respect to universal norms, and relates it to previous height pairings for abelian varieties and for Galois representations satisfying Panchishkin’s condition.
For the entire collection see [Zbl 0801.00020].

##### MSC:
 11G35 Varieties over global fields 14F30 $$p$$-adic cohomology, crystalline cohomology 11G09 Drinfel’d modules; higher-dimensional motives, etc.