\(p\)-adic heights (Q2706483)

The study of local height pairings on abelian varieties (over local fields) was begun by \textit{A. Néron} [Ann. Math. (2) 82, 249--331 (1965; Zbl 0163.15205)], and a canonical global height pairing for abelian varieties (over global fields) was defined independently by Néron and Tate. These pairings took values in archimedean fields, but \textit{J. Oesterlé} [Théorie des Nombres, Sémin. Delange-Pisot-Poitou, Paris 1980--81, Prog. Math. 22, 175--192 (1982; Zbl 0531.14016)] gave analogues (depending on an additional choice) with values in \(\mathbb Q_p\), basing his construction on an equivalent definition of the archimedean pairings due to \textit{S. Bloch} [Invent. Math. 58, 65--76 (1980; Zbl 0444.14015)]. NEWLINENEWLINEThe authors' goal in the present paper is to repeat the constructions in Oesterlé's paper, but for a general ``\(p\)-adic quasicharacter'' in the sense of \textit{Yu. I. Manin} [Math. USSR, Sb. 12 (1970), 325--342 (1971); translation from Mat. Sb., N. Ser. 83(125), 331--348 (1970; Zbl 0214.48402)] and \textit{Yu. G. Zarkhin} [Math. USSR, Izv. 6 (1972), 491--503 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 497--509 (1972; Zbl 0244.14009)]. Oesterlé had used the particular quasicharacters related to the norm. NEWLINENEWLINEThe reviewer is confused by several things in the current paper. First, Néron models are mentioned on pages 38 and 39, but it seems that only their generic fibers are used. Second, there is a condition in theorem 1.5 that the homomorphism \(c_v\colon k_v^* \rightarrow \mathbb Q_p\) be trivial on the values of the Weil pairing between the torsion points of \(A(k_v)\) and \(A'(k_v)\); this condition, apparently from Zarkhin's paper on quasicharacters taking values in general injective groups, is vacuous here, since \(\mathbb Q_p\) is torsion-free. Third (and most seriously), the domain of \(\tilde{c}_v\) depends on \(\Delta\), so one has the task of choosing a \(\tilde{c}_v\) for each \(\Delta\) in a way that is compatible with addition of divisors \(\Delta\), if one wants the pairing in theorem 1.5 to be additive in \(\Delta\). NEWLINENEWLINENEWLINEFinally, it should be mentioned that for certain abelian varieties there exist definitions of canonical global \(p\)-adic height pairings, not suffering from the ambiguity of a choice of \(\widetilde{c}_v\), see \textit{P. Schneider} [Invent. Math. 69, 401-409 (1982; Zbl 0509.14028)] and \textit{B. Mazur} and \textit{J. Tate} [see Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 195-237 (1983; Zbl 0574.14036)], and references cited there.