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.

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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In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on GL_2) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor C(π×π_{χ^{−1}}). As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

For an odd prime p, we prove the p-part of the Birch and Swinnerton-Dyer formula for semistable elliptic curves of analytic rank one.