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Selmer groups and Heegner points in anticyclotomic \(\mathbb{Z}_ p\)-extensions. (English) Zbl 0862.11043
Let \(E\) be a modular elliptic curve, and let \(p\) be a prime of good ordinary reduction for \(E\). Fix an imaginary quadratic field \(K\) which satisfies the Heegner hypothesis. Write \(K_\infty\) for the anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\), and \(\Lambda \) for the Iwasawa algebra relative to the extension \(K_\infty/K\). There is a collection of Heegner points defined over \(K_\infty\) coming from a modular parametrization of \(E\). Assume that one of these points has infinite order. (It is expected that this condition always holds.)
One result of the paper states that the \(p^\infty\)-Selmer group \(\text{Sel}_{p^\infty} (E/K_\infty)\) of \(E\) over \(K_\infty\) has \(\Lambda\)-corank equal to one. (This equality has been conjectured by B. Mazur.) Moreover, it is shown that the \(\Lambda\)-cotorsion quotient of \(\text{Sel}_{p^\infty} (E/K_\infty)\) is annihilated by an element \(\rho_\infty\) of \(\Lambda\), which encodes the way the Heegner points “sit inside” the Selmer group of \(E/K_\infty\). This proves a weak version of a conjecture of B. Perrin-Riou, which can be viewed as the analogue in the present context of the main conjecture of Iwasawa theory for cyclotomic fields. The results of the paper are obtained by building on the methods of V. A. Kolyvagin and using techniques of Iwasawa theory.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14H52 Elliptic curves
11G05 Elliptic curves over global fields
11R23 Iwasawa theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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