Search

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 C each of these Calabi–Yau threefolds Y is 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 X to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi–Yau variety over C . [ABSTRACT FROM AUTHOR]

Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields. Their approach using A 1 -enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the 28 bitangents to a smooth plane quartic. However, it turns out that the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed "line at infinity," which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line. [ABSTRACT FROM AUTHOR]

Inspired by the analogy between the group of units F p × of the finite field with p elements and the group of points E (F p) of an elliptic curve E / F p , 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 ≤ x τ (p + a) ∼ C x . 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 C above. 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. [ABSTRACT FROM AUTHOR]

In this note we compute the number of general points through which a general Brill-Noether space curve passes. [ABSTRACT FROM AUTHOR]

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