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This paper proves Birch and Swinnerton-Dyer conjecture for Q-curves defined over real quadratic number field with everywhere good reductions.

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Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [$Acta Math$., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.