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.

Let π be a cuspidal automorphic representation of PGL_2(AQ) of arithmetic conductor C and archimedean parameter T, and let ϕ be an L2-normalized automorphic form in the space of π. The sup-norm problem asks for bounds on ∥ϕ∥_∞ in terms of C and T. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L2-mass |ϕ|^2(g)dg of ϕ. Previous work on these problems in the conductor-aspect has focused on the case that ϕ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C, the local component π_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms ϕ for which the local components ϕ_p∈π_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL_2(R). For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if π_∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C^{1/8−ϵ} k^{1/4−ϵ} ≪_ϵ |ϕ|_∞ ≪_ϵ C^{1/8+ϵ} k^{1/4+ϵ}. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

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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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