Canonical height pairings via biextensions.

*(English)*Zbl 0574.14036
Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 195-237 (1983).

[For the entire collection see Zbl 0518.00004.]

Let K be a global field, A an abelian variety defined over K, and A’ the dual variety. For each \({\mathbb{Z}}_ p\)-extension of K with finitely many ramified primes that are also primes of ordinary reduction for A, the authors construct a \({\mathbb{Q}}_ p\)-valued height pairing on A(K)\(\times A'(K)\). The existence of several p-adic height pairings is in marked contrast to the case of the canonical \({\mathbb{R}}\)-valued height pairing, which is, of cource, unique. A p-adic valued height pairing has also been constructed by P. Schneider [”p-adic height pairings. I”, Invent. Math. 69, 401-409 (1982; Zbl 0509.14048)] for the cyclotomic \({\mathbb{Z}}_ p\)-extension when A has good, ordinary reduction at the primes of K dividing p. Under these conditions on A the authors prove that their p- adic pairing for the cyclotomic \({\mathbb{Z}}_ p\)-extension agrees with Schneider’s pairing. Both Schneider’s pairing and the authors’ are modelled after Bloch’s definition (using local splittings) of the classical archimedean height. The authors use the canonical biextension of (A,A’) by \({\mathbb{G}}_ m\), and develop a theory of local splittings of such biextensions. Schneider, on the other hand, obtains his local splittings by studying local universal norms. In order to prove that their pairing agrees with Schneider’s the authors verify that the local splittings used to construct the different pairings are actually the same. Unfortunately, neither Schneider nor the authors are able to prove that the pairings are non-degenerate.

More recently [part II of his cited paper, Invent. Math. 79, 329-374 (1985; Zbl 0571.14021)], P. Schneider has extended his pairing to a pairing on the p-Selmer groups of A and A’. One obtains the original p- adic pairing by restricting to points. Other p-adic pairings have been constructed by Perrin-Riou for elliptic curves with complex multiplication, and by Néron.

Let K be a global field, A an abelian variety defined over K, and A’ the dual variety. For each \({\mathbb{Z}}_ p\)-extension of K with finitely many ramified primes that are also primes of ordinary reduction for A, the authors construct a \({\mathbb{Q}}_ p\)-valued height pairing on A(K)\(\times A'(K)\). The existence of several p-adic height pairings is in marked contrast to the case of the canonical \({\mathbb{R}}\)-valued height pairing, which is, of cource, unique. A p-adic valued height pairing has also been constructed by P. Schneider [”p-adic height pairings. I”, Invent. Math. 69, 401-409 (1982; Zbl 0509.14048)] for the cyclotomic \({\mathbb{Z}}_ p\)-extension when A has good, ordinary reduction at the primes of K dividing p. Under these conditions on A the authors prove that their p- adic pairing for the cyclotomic \({\mathbb{Z}}_ p\)-extension agrees with Schneider’s pairing. Both Schneider’s pairing and the authors’ are modelled after Bloch’s definition (using local splittings) of the classical archimedean height. The authors use the canonical biextension of (A,A’) by \({\mathbb{G}}_ m\), and develop a theory of local splittings of such biextensions. Schneider, on the other hand, obtains his local splittings by studying local universal norms. In order to prove that their pairing agrees with Schneider’s the authors verify that the local splittings used to construct the different pairings are actually the same. Unfortunately, neither Schneider nor the authors are able to prove that the pairings are non-degenerate.

More recently [part II of his cited paper, Invent. Math. 79, 329-374 (1985; Zbl 0571.14021)], P. Schneider has extended his pairing to a pairing on the p-Selmer groups of A and A’. One obtains the original p- adic pairing by restricting to points. Other p-adic pairings have been constructed by Perrin-Riou for elliptic curves with complex multiplication, and by Néron.

Reviewer: S.Kamienny

##### MSC:

14K15 | Arithmetic ground fields for abelian varieties |

14G25 | Global ground fields in algebraic geometry |