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The universal vectorial bi-extension and $$p$$-adic heights. (English) Zbl 0763.14009
This paper is a sequel to the author’s joint paper with B. H. Gross in Algebraic number theory — in Honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-functions, Berkeley/CA 1987, Adv. Stud. Pure Math. 17, 73-81 (1989; Zbl 0758.14009), abbreviated by [CG]. There are several definitions of $$p$$-adic heights on a curve $$C$$ defined over an algebraic number field $$K$$, namely by B. Mazur and J. Tate in Arithmetic and geometry, Pap. dedic. Shafarevich, Vol. I; Arithmetic, Prog. Math. 35, 195-237 (1983; Zbl 0574.14036), abbreviated by [MT], by P. Schneider [Invent. Math. 69, 401-409 (1982; Zbl 0509.14048 and 79, 329-374 (1985; Zbl 0571.14021)] and in [CG]. In the paper under review, there is given a proof that [MT] and [CG] define the same $$p$$- adic height.
Let $$A/S$$ be an abelian scheme (e.g. the Jacobian variety of $$C)$$, then there exists a canonical bi-extension $$B$$ of $$\hat A\times A$$. The splitting of this bi-extension is used in [MT] for the construction of the $$p$$-adic height. Let $$G$$ and $$\hat G$$ denote the universal vectorial extensions of $$A$$ and $$\hat A$$. The pullback $$B^ \#$$ of $$B$$ is called the universal vectorial bi-extension of $$\hat G\times G$$. It carries a canonical differential $$\eta$$ such that its exterior derivative “is” the pairing between the first de Rham cohomology of $$\hat A$$ and $$A$$. The pair $$(B^ \#,\eta)$$ provides a one-to-one correspondence between formal splittings of $$\hat G\to\hat A$$, $$G\to A$$ and “splittings” of $$B$$. The splittingof $$G\to A$$ is equivalent to a splitting of the de Rham cohomology of $$A$$. The latter splitting is the key in the construction of [CG]. The paper also seems to review the construction of $$p$$-adic height given in [CG].

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G20 Local ground fields in algebraic geometry
##### Keywords:
$$p$$-adic height on a curve
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##### References:
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