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Congruences and rationality of Stark-Heegner points. (English) Zbl 1308.11063
Summary: Let \(A/\mathbb Q\) be a modular abelian variety attached to a weight \(2\) new modular form of level \(N=pM\), where \(p\) is a prime and \(M\) is an integer prime to \(p\). When \(K/\mathbb Q\) is an imaginary quadratic extension the Heegner points, that are defined over the ring class fields \(H/K\), can contribute to the growth of the rank of the Selmer groups over \(H\). When \(K/\mathbb Q\) is a real quadratic field the theory of Stark-Heegner points provides a conjectural explanation of the growth of these ranks under suitable sign conditions on the \(L\)-function of \(f/K\). The main result of the paper relates the growth of the Selmer groups to the conjectured rationality of the Stark-Heegner points over the expected field of definition.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Holomorphic modular forms of integral weight
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
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