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Derived \(p\)-adic heights. (English) Zbl 0882.11036
Let \(E\) be an elliptic curve defined over a number field \(K\). Let \(K_\infty/K\) be a \(\mathbb{Z}_p\)-extension of \(K\); put \(\Gamma=\text{Gal}(K_\infty,K)\). Denote by \(\Lambda=\mathbb{Z}_p[[\Gamma]]\) the Iwasawa algebra. Given any topological generator \(\gamma\) of \(\Gamma\), write \(I=(\gamma- 1)\Lambda\). Assume that \(E\) has good reduction above a prime \(p\). Let \(\overline S^{(1)}_p=\displaystyle{\varprojlim_n} \text{Sel}_{p^n}(E/K)\) denote the inverse limit of the \(p^n\)-Selmer groups with respect to the multiplication by \(p\) maps, and for \(k\geq 2\) let \(\overline S^{(k)}_p\) denote the null-space of \(\langle\langle , \rangle\rangle_{k-1}\). The first result (Theorem 2.18) states the existence, for \(1\leq k\leq p-1\), of a sequence of canonical pairings (derived \(p\)-adic heights) \[ \langle\langle , \rangle\rangle_k:\overline S^{(k)}_p\times\overline S^{(k)}_p\to I^k/I^{k+1}\otimes\mathbb{Q}. \] The authors show (Theorem 2.8) that the restriction of \(\langle\langle , \rangle\rangle_1\) to \(E(K)_p\) is equal to the \(p\)-adic height pairing defined by Mazur-Tate and Schneider. In general, these pairings are either symmetric or alternating, depending on whether \(k\) is odd or even (Theorem 2.7). Theorems 2.18 and 2.7 give a description of \(\overline S^{(k)}_p\) in terms of the \(\Lambda\)-module structure of the Selmer group of \(E/K_\infty\).

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11R23 Iwasawa theory
14G20 Local ground fields in algebraic geometry
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