No Arabic abstract
Let $k$ be a number field. We give an explicit bound, depending only on $[k:mathbf{Q}]$ and the discriminant of the N{e}ron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on the Generalised Riemann Hypothesis, we also give an explicit bound, depending only on $[k:mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. In addition, we show how to obtain a bound, depending only on $[k:mathbf{Q}]$, on the number of $mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $text{Br}, Y/ text{Br}_1, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Neron-Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd-torsion of $text{Br}, Y / text{Br}_1, Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Neron-Severi lattice, $(text{Br}, Y / text{Br}_1, Y)[p^infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant.
We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $mu_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.
We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.
In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $mathbf{A}_1+mathbf{A}_3$ and prove an analogue of Manins conjecture for integral points with respect to its singularities and its lines.
We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special cycles of virtual dimension 0 on the moduli stack of unitary shtukas with r legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.