Congruences and rationality of Stark–Heegner points

Let A/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/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/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.