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AbstractIn this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed.

A Correction to this paper has been published: https://doi.org/10.1007/s40993-021-00307-4

In this paper we investigate the Q-rational points of a class of simply connected Calabi–Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer–Manin obstruction to weak approximation. Hosono and Takagi showed that over Ceach of these Calabi–Yau threefolds Yis derived equivalent to a Reye congruence Calabi–Yau threefold X. We show that these derived equivalences may also be constructed over Q, and we give sufficient conditions for Xto not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi–Yau variety over C.

Inspired by the analogy between the group of units Fp×of the finite field with pelements and the group of points E(Fp)of an elliptic curve E/Fp, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum ∑p≤xτ(p+a)∼Cx. In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums in place of the constant Cabove. Specifically, by analyzing the division fields of an elliptic curve E/Q, we prove bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves E/Q. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.