Kolyvagin’s descent and Mordell-Weil groups over ring class fields. (English) Zbl 0712.14008

Bertolini, Massimo; Darmon, Henri

J. Reine Angew. Math. 412, 63-74 (1990).

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.

Reviewer: M.Bertolini; H.Darmon

MSC:

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

Keywords:

Kolyvagin’s descent; Mordell-Weil groups; modular elliptic curve; ring class field extension; Heegner point; Birch and Swinnerton-Dyer conjecture