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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
##### Keywords:
derived $$p$$-adic heights; Selmer groups; Iwasawa algebra
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