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

Abstract

Let N ≡ 1(mod 4) be a square-free positive integer, let e be the primitive quadratic character of conductor N , and let f ∈ S2(Γ0(N), e) be a newform with fourier coefficients in a quadratic field. Shimura associates to f an elliptic curve E defined over the real quadratic field F = Q( √ N) which is isogenous over F to its Galois conjugate. Let M/F be a quadratic extension of F which is neither CM or totally real, and denote by EM the twist of E with respect to M/F . The first main result of this article is that, if L′(EM/F, 1) 6= 0, then EM (F ) has rank one and LLI(E/F ) is finite. The proof rests on a fundamental result of Yuan, Zhang and Zhang and on an explicit Heegner point in EM (F ) arising from the modular parametrization X1(N) → E. We formulate a conjecture relating this Heegner point to Stark-Heegner points arising from ATR cycles of real dimension one on Hilbert modular surfaces, and present some numerical evidence for this conjecture.