The Birch and Swinnerton-Dyer conjecture for ℚ-curves and Oda's period relations

Let N ≡ 1(mod 4) be a square-free positive integer, let ε be the primitive quadratic character of conductor N, and let f ∈ S₂(Γ₀(N), ε) be a newform with fourier coefficients in a quadratic field. Shimura associates to f an elliptic curve E defined over the real quadratic field which is isogenous over F to its Galois conjugate. Let M/F be a quadratic extension of F which is neither CM nor totally real, and denote by E_M the twist of E with respect to M/F. The first main result of this article is that, if L′(E_M/F, 1) ≠ 0, then E_M(F) has rank one and III(E/F) is finite. The proof rests on a fundamental result⁴² of Yuan, Zhang and Zhang and on an explicit Heegner point in E_M(F) arising from the modular parametrization X₁(N) → E. We formulate a conjecture relating this Heegner point to Stark-Heegner points⁸ arising from ATR cycles of real dimension one on Hilbert modular surfaces, and present some numerical evidence for this conjecture.