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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.

In the classical theory of multiple zeta values (MZV's), Furusho proposed a conjecture asserting that the p-adic MZV's satisfy the same Q-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho's conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations.

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.

In 1962 Sprind\v{z}uk proved Mahler's conjecture in both the real and complex cases. Baker gave a generalized result by using a modified version of Sprind\v{z}uk's method. Later, Baker and Schmidt derived the Hausdorff dimensions of sets which are defiend in terms of approximation by algebraic numbers of bounded degrees by using Baker's theorem. In this article we will prove two analogue theorems in the fields of formal power series over finite finite fields.