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Kolyvagin’s descent and Mordell-Weil groups over ring class fields. (English) Zbl 0712.14008
The authors prove the following theorem: Let E be a modular elliptic curve and H/K be a ring class field extension of a suitable imaginary quadratic field K. Let \(\alpha\in E(H)\) denote a Heegner point defined over H, and let \(\chi\) denote a complex character of the Galois group of H/K. - If the \(\chi\)-component of \(\alpha\) is nonzero, then the complex dimension of the \(\chi\)-component of E(H) is one.
This statement was conjectured by B. H. Gross in Modular forms, Symp. Durham/Engl. 1983, 87-105 (1984; Zbl 0559.14011), in view of the Birch and Swinnerton-Dyer conjecture and the Gross-Zagier theorem, and proved by V. A. Kolyvagin [“Euler systems” in the Grothendieck Festschrift, Vol. II, Proc. Math. 87, 435-483 (1990)] when \(\chi\) is the trivial character. The authors apply the ideas of Kolyvagin to prove the general case.

14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
14G35 Modular and Shimura varieties
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