Iwasawa theory of quadratic twists of X0(49)

The field (K = mathbb{Q}left( {sqrt { - 7} } right)) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X ₀(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X ₀(49) by the quadratic extension (KK(sqrt M )/K), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F _∞ = K(E _p∞), where E _p∞ denotes the group of p^∞-division points on E. Moreover, writing B for the twist of X ₀(49) by (K(sqrt[4]{{ - 7}})/K), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z₂-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.