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

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.

The main results of this paper involve general algebraic differentials ω on a general pencil of algebraic curves. We show how to determine whether ω is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations A^2−DB^2=1 over a polynomial ring. We determine whether the Pell equation (with squarefree D) is solvable for infinitely many members of the pencil.

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