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Abelian varieties, \(p\)-adic heights and derivatives. (English) Zbl 0829.11034

Frey, Gerhard (ed.) et al., Algebra and number theory. Proceedings of a conference held at the Institute of Experimental Mathematics, University of Essen, Germany, December 2-4, 1992. Berlin: de Gruyter. 247-266 (1994).
Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with good ordinary reduction at a prime \(p\) and let \(T_p (E)\) be the \(p\)-adic Tate module of \(E\). Let \(\mathbb{Q}_n\) be the \(n\)-th level of the cyclotomic \(\mathbb{Z}_p\)-extension of \(\mathbb{Q}\). Let \(\omega\) be a holomorphic differential on \(E\) and let \(z = \{z_n\} \in \varprojlim H^1 (\mathbb{Q}_n, T_p (E))\). The author defines an Iwasawa function \[ L_{z, \omega} : \operatorname{Hom}_{\text{cont}} \bigl (\text{Gal} (\mathbb{Q}_n/ \mathbb{Q}), \mathbb{Z}_p^\times \bigr) \to \mathbb{Q}_p \] and proves the following: Let 1 denote the trivial character. (1) If \(L_{z, \omega} (\text{\textbf{1}})\neq 0\) then \(E (\mathbb{Q})\) is finite. (2) If \(L_{z, \omega} (\text{\textbf{1}})= 0\) and the \(p\)-part of the Tate-Shafarevich group is finite, then \(z_0 \in E (\mathbb{Q}) \otimes \mathbb{Z}_p\). Moreover, in case (2), a certain \(p\)-adic height pairing \(\langle z_0, x \rangle\) can be expressed as a multiple of \(L_{z, \omega} '(\text{\textbf{1}})\log_\omega (x)\) for all \(x \in E(\mathbb{Q}) \otimes \mathbb{Z}_p\), where \(\log_\omega\) is the \(p\)-adic logarithm corresponding to \(\omega\). When \(E\) has complex multiplication, the author [Invent. Math. 107, 323-350 (1992; Zbl 0770.11033)] constructed \(\{z_n\}\) such that \(L_{z, \omega}\) is essentially the \(p\)-adic \(L\)-function of \(E\). When \(E\) is a modular elliptic curve, recent unpublished work of K. Kato gives classes \(z_n\) for each \(n\), and it is suspected that these classes give an element of the inverse limit and yield the \(p\)-adic \(L\)-function. As a corollary to the above theorem, when \(E\) has Mordell-Weil rank one, the author shows how to construct a rational point of infinite order. This was previously done by the author [ibid.] for CM curves. The proof uses Tate duality and the height pairing of B. Perrin-Riou [Invent. Math. 109, 137-185 (1992; Zbl 0781.14013)] to prove a general theorem for abelian varieties that specializes to the result for elliptic curves over \(\mathbb{Q}\). B. Perrin-Riou [Ann. Inst. Fourier 43, No. 4, 945-995 (1993)] has independently obtained the results of this paper.
For the entire collection see [Zbl 0793.00015].

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
14G20 Local ground fields in algebraic geometry
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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