For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field. He proved these conjectures for low dimensions. We prove Kato's conjecture over number fields. In particular this gives a Hasse principle for function fields F over a number field K, involving the corresponding function fields over all completions of K. We get the same results over global fields K of positive characteristic, for coefficients invertible in K. This was proved earlier by M. Kerz and S. Saito, by another method. Finally we obtain a conjecture of Kato over a finite field, and a generalization to finitely generated fields K, assuming resolution of singularities or that the coefficents are invertible in K. The latter case was again obtained earlier by M. Kerz and S. Saito, by different methods.
We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric Q over a number field F. Besides, we show that such summands are twists of Rost motives. In the case when F has at most one real embedding we describe a complete motivic decomposition of Q.
We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.
We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.
We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.
We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bretèche and Browning.
We prove the Hasse principle for a smooth projective variety $X\subset \PP^{n-1}_\Q$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\Q$.
Let $k$ be a global field and let $L_0$,...,$L_m$ be finite separable field extensions of $k$. In this paper, we are interested in the Hasse principle for the multinorm equation $\underset{i=0}{\overset{m}{\prod}}N_{L_i/k}(t_i)=c$. Under the assumption that $L_0$ is a cyclic extension, we give an explicit description of the Brauer-Manin obstruction to the Hasse principle. We also give a complete criterion for the Hasse principle for multinorm equations to hold when $L_0$ is a meta-cyclic extension.
The aim of this note is to revisit the question of local-global principles for embeddings of etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in a joint paper with Tingyu Lee and Raman Parimala. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise independent field extensions, then the Hasse principle holds.
In this paper, we consider the following problem: Does there exist a cubic surface over $\mathbb{Q}$ which contains no line over $\mathbb{Q}$, yet contains a line over every completion of $\mathbb{Q}$? This question may be interpreted as asking whether the Hilbert scheme of lines on a cubic surface can fail the Hasse principle. We also consider analogous problems, over arbitrary number fields, for other del Pezzo surfaces and complete intersections of two quadrics.
Let $X$ be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for $X$. We prove that if the generic fiber has a zero-cycle of degree $d$ over the generic point, then the Brauer classes whose orders are prime to $d$ do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, ce…
Cubic surfaces failing the integral Hasse principle
University of Bath（巴斯大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$. Hoshi, Kanai and Yamasaki [HKY22] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\leq 15$ and $n\neq 12$. In this paper, we determine $64$ cases with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ and give a necessary and sufficient condition for the Hasse…
Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain $\mathcal{O}_{\{\infty\}} = \mathbb{F}_q[C^\text{af}]$ of $K$, over which we consider a non-degenerate bilinear and symmetric form $f$ with orthogonal group $\underline{\textbf{O}}_V$. We show that the set $\text{Cl}_\infty(\underline{\textbf{O}}_V)$ of $\mathcal{O}_{\{\infty\}}$-isomorphism classes in the genus of $f$ of rank $n>2$, is bijective as a pointed set to …
We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to $\mathbb{F}_p^{\oplus r}$ ($r \leq 2$), we show that the product of the restriction maps for places in a set of places $S$ is injective if the Dirichlet density of $S$ is strictly larger than $1 - p^{-r}$. We give applications to the local-global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.
We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for homogeneous spaces with nilpotent stabilizer. We thus generalize recent results by Harpaz and Wittenberg on finite "hyper-solvable" stabilizers. In particular, this result is true for abelian stabilizers without any hypotheses on the ambient group, which generalizes classic results by Borovoi on the subject. We prove moreover that these spaces have the real approximation property. On montre que l'obstruction de Brauer-Manin est la seule obstruction au principe de Hasse pour les espaces homogènes à sta…
Soit $K$ un corps global et $G$ un $K$-groupe fini résoluble. Sous certaines hypothèses sur une extension déployant $G$, on démontre que l'espace homogène $V:=G'/G$ avec $G'$ un $K$-groupe semi-simple simplement connexe vérifie l'approximation faible. On utilise une version plus précise de ce résultat pour démontrer le principe de Hasse pour des espaces homogènes $X$ sous un $K$-groupe $G'$ semi-simple simplement connexe à stabilisateur géométrique $\bar G$ fini et résoluble, sous certaines hypothèses sur le $K$-lien $(\bar G,κ)$ défini par $X$. ----- Let $K$ be a global field and $G$ a finite…
Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erdős.
We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $α$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, α)$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra represen…
The aim of this paper is to revisit the question of local-global principles for embeddings of étale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in [BLP 18]. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise linearly disjoint field extensions, then the Hasse principle holds, and that if an embedding exists after an odd degree extension, then it also exists over the global field itself. …
We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$, and smooth complete intersections of two quadrics of dimension at least $3$ in odd characteristics. In Appendix A we explain how to modify a previous argument of the author to prove weak approximation for cubic hypersurfaces defined over function fields of curves over algebraically closed fields of characteristic at least $7$. In Appendix B we…
Nous montrons, pour une grande famille de propriétés $P$ des espaces homogènes, que $P$ vaut pour tout espace homogène d'un groupe linéaire connexe dès qu'elle vaut pour les espaces homogènes de $\mathrm{SL}_n$ à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d'une importante conjecture de Colliot-Thélène sur l'obstruction de Brauer-Manin au principe de Hasse et à l'approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s'applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-cycles. We pr…
In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.
