Tony Quertier
We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over the reals. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists, and if so, to compute one.
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Failures of the Integral Hasse Principle for Affine Quadric Surfaces
本文研究了在整数定义的仿射二次曲面上，积分哈斯原理失效的频率及反例出现情况。
Hasse principle for intersections of two quadrics via Kummer surfaces
通过Kummer曲面证明亏格2曲线雅可比簇2-覆盖的Hasse原理新情形，并推导四次del Pezzo曲面及高维射影空间中两个二次曲面光滑完全交点的Hasse原理。
Hasse principle for the Chow groups on quadric fibrations
本文研究了数域上 2 或 3 维二次纤维化的零循环相对 Chow 群，给出了其整体到局部映射为单射的充分条件。
This paper proves the Hasse principle and weak approximation for varieties defined by the smooth intersection of three quadratics in at least 19 variables, over arbitrary number fields.
The analytic Hasse Principle for certain singular intersections of quadrics in $\mathbb{P}^9$
本文利用自守方法，证明了在特定奇点条件下至少含10个变量的有理系数二次型对公共零点个数的渐近公式，推广了 Browning 与 Munshi 的工作。
It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.
Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the Gaussians.
We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
On the arithmetic of intersections of two quadrics containing a conic
本文重录1993年未发表手稿，证明了Colliot-Thélène、Sansuc与Swinnerton-Dyer关于含圆锥曲线的两个二次曲面交算术性质的两个猜想。
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…
Complete intersections of cubic and quadric hypersurfaces over $\mathbb{F}_q(t)$
本文用二维 delta 方法证明 $\mathbb{F}_q(t)$ 上高维非奇异三次与二次超曲面完全交的有界高度有理点渐近公式，并验证弱逼近。
For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, we show that the Brauer-Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics, which is a consequence of recent advances in additive combinatorics.
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.
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 $\Q$.
For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.
For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.
We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent arguments.
Admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le même résultat vaut pour n=4, c'est-à-dire pour les surfaces de del Pezzo de degré 4, lorsque le groupe de Brauer est réduit aux constantes et que la surface est suffisamment générale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics i…
Strong approximation for the intersection of two quadrics
本文研究两个仿射二次曲面交集的强逼近问题，并将其应用于证明秩四数域上非分裂纤维由二次扩张分裂情形下的弱逼近纤维化方法。
When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero over $\mathbb Q$. We also show that a positive proportion of all ternary cubic forms over $\mathbb Z$ nontrivially satisfy the Hasse principle, i.e., they possess a zero over every completion of $\mathbb Q$ and also possess a zero over $\mathbb Q$. Analogous results are proven for other genus one models, namely, for equations of the form $z^2=f(x,y)$ where $f$ …
Quadratic points on intersections of two quadrics
该研究证明数域上维数≥2的两个二次曲面的光滑完全交指数整除2，即存在次数为2的有理0-闭链，推进了代数数论与代数几何领域相关问题的研究。
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a par…
Solvable points on intersections of quadrics, cubics, and quartics
本文建立多项式下界，使得在二次、三次和四次超曲面交线X上存在稠密的可解点集，方法是将极超曲面经典理论与X上j维线性子空间的Fano簇联系起来。
A Kloosterman refinement for function fields $K=\mathbb{F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n-1}_{K}$ , under the assumption that $q$ is odd and $n\geq 9$.
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics…
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops further the technique initiated by Swinnerton-Dyer in 1993, and later generalised by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer, for studying rational points on pencils of curves of genus 1. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and to higher-dimensional intersections of two quadrics. Conditionally on two wel…
Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov
本文综合 Lichtenbaum 与 Salberger 的结果，推广两个二次曲面交上二次点存在性的结论，并重新证明或另证 Heath-Brown 及 Iyer–Parimala 的 Hasse 与局部—整体原理定理。
We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in 8 variables. The argument develops work of Colliot-Thelene, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off 3 hyperbolic planes.
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to be $C_0^\infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.
Period -Index problem for hyperelliptic curves
本文研究双曲曲线的周期-指数问题，证明了2- torsion元素中指数与幂次相等，并通过同构建立与二次曲面 pencils 的关系，推导出弱Hasse原理。
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