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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
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