Manin's conjecture for a cubic surface with D_5 singularity

Manin猜想在弦三次四重流形上的成立

Manin's conjecture for the chordal cubic fourfold

本文证明弦三次四重流形上Manin猜想的薄集版本，通过将其与射影平面的对称平方同构，归结为Schmidt的二次点结果。

Manin's conjecture for a certain singular cubic surface

We prove Manin's conjecture for a singular cubic surface S with a singularity of type E6. If U is the open subset of S obtained by deleting the unique line from S, then the number of rational points in U with anticanonical height bounded by B behaves asymptotically as cB(log B)^6, where the constant c agrees with the one conjectured by Peyre.

On Manin's conjecture for a certain singular cubic surface

Let U denote the open subset formed by deleting the unique line from the singular cubic surface x_1x_2^2+x_2x_0^2+x_3^3=0. In this paper an asymptotic formula is obtained for the number of rational points on U of bounded height, which thereby verifies the Manin conjecture for this particular surface.

Manin's conjecture for a cubic surface with 2A_2+A_1 singularity type

We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath-Brown) and draws on the work of Deligne.

Manin's conjecture for two quartic del Pezzo surfaces with 3A_1 and A_1+A_2 singularity types

We prove Manin's conjecture for two del Pezzo surfaces of degree four which are split over Q and whose singularity types are respectively 3A_1 and A_1+A_2. For this, we study a certain restricted divisor function and use a result about the equidistribution of its values in arithmetic progressions. In this task, Weil's bound for Kloosterman sums plays a key role.

Persistence of the Brauer-Manin obstruction on cubic surfaces

Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3\nmid[L:k]$. Therefore, the conjecture of Colliot-Thélène and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement…

On the Brauer-Manin obstruction for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This covers the vast majority of the cases when this group is non-zero. Our approach is to associate a Brauer class with every Galois invariant double-six. We show that all order two Brauer classes may be obtained in this way. We also recover Sir Peter Swinnerton-Dyer's result that $\Br(S)/\Br(\bbQ)$ may take only five values.

Markoff型三次曲面上整点的Brauer-Manin障碍

Brauer-Manin obstruction for integral points on Markoff-type cubic surfaces

该研究延续相关工作，探讨Markoff型三次曲面整点的Brauer-Manin障碍，构造可由该障碍解释的强逼近反例并给出计数结果，还提出无法用代数Brauer-Manin障碍解释的整点Hasse原理反例。

Rational Points on the Fermat Cubic Surface

We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.

Counting rational points over number fields on a singular cubic surface

A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases of Manin's conjecture over the field Q of rational numbers. Combining this method with techniques developed by Schanuel, we give a proof of Manin's conjecture over arbitrary number fields for the singular cubic surface S given by the equation w^3 = x y z.

关于 Birch 和 Swinnerton-Dyer 的三次曲面

On Birch and Swinnerton-Dyer's cubic surfaces

本文推广了 Birch 和 Swinnerton-Dyer 1975 年的工作，利用现代计算机代数软件证明了更大范围的三次曲面存在 Brauer--Manin 障碍以阻碍 Hasse 原理，从而对无限多个三次曲面验证了 Colliot-Thélène--Sansuc 猜想。

Manin's conjecture for a class of singular cubic hypersurfaces

Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two aspects: first, it can be viewed as a modest start on the study of density of rational points on those singular cubic hypersurfaces which are not covered by the classical theorems of Davenport or Heath-Brown; second, it proves Manin's conjecture for singular cubic hypersurfaces $S_n$ defined above.

Manin's conjecture for singular cubic hypersurfaces

Let S Q denote x 3 = Q(y 1 ,. .. , y m)z where Q is a primitive positive definite quadratic form in m variables with integer coefficients. This S Q ranges over a class of singular cubic hypersurfaces as Q varies. For S Q we prove (i) Manin's conjecture is true if Q is locally determined with 2 | m and m 4; (ii) in general Manin's conjecture is true up to a leading constant if 2 | m and m 6.

Gaussian rational points on a singular cubic surface

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xy…

On the Brauer group of diagonal cubic surfaces

We are concerned with finding explicit generators of the Brauer group of diagonal cubic surfaces in terms of norm residue symbols, which was originally studied by Manin. We introduce the notion of uniform generators and find that the Brauer group of some classes of diagonal cubic surfaces have uniform generators. However, we also prove that the Brauer group of general diagonal cubic surfaces do not have such ones. This reveals that a result of Manin for certain diagonal cubic surfaces cannot be generalized in some sense.

On a senary cubic form

A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.

三个 del Pezzo 曲面的 Manin-Peyre 猜想

The Manin-Peyre conjecture for three del Pezzo surfaces

本文证明了三个奇异 del Pezzo 曲面的 Manin-Peyre 猜想，采用统一的新方法改进了误差项，并指出该方法可推广至大多数次数不低于 3 的奇异 del Pezzo 曲面。

Counting rational points on smooth cubic surfaces

We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.

Quadratic congruences on average and rational points on cubic surfaces

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.

Tamagawa numbers of diagonal cubic surfaces of higher rank

We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previo…

Manin's Conjecture for a Singular Sextic del Pezzo Surface

We prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square

We complete the study of points of bounded height on irreducible non-normal cubic surfaces by doing the point count on the cubic surface $W$ given by $t_0^2 t_2 = t_1^2 t_3$ over any number field. We show that the order of growth agrees with a conjecture by Batyrev and Manin and that the constant reflects the geometry of the variety as predicted by a conjecture of Batyrev and Tschinkel. We then provide the point count for its symmetric square $\mathrm{Sym}^2 W$. Although we can explain the main term of the counting function, the Batyrev--Manin conjecture is only satisfied after removing a thin…

The power-saving Manin-Peyre's conjectures for a senary cubic

Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with bihomogeneous coordinates $[x_1:x_2:x_3],[y_1:y_2,y_3]$ and in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with multihomogeneous coordinates $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ defined by the same equation $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$. We thus improve on recent work of Blomer, Brüdern and Salberger \cite{BBS} and provide a different proof based on…

The density of rational points on a certain singular cubic surface

We show that the number of non-trivial rational points of height at most $B$, that lie on the cubic surface $x_1x_2x_3=x_4(x_1+x_2+x_3)^2$, has order of magnitude $B(\log B)^6$. This agrees with the Manin conjecture.

Affine congruences and rational points on a certain cubic surface

We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of Browning and answers a problem posed by Tschinkel.

Manin's conjecture for a singular quartic del Pezzo surface

We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type $2\Aone$ and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics, we prove a result of independent interest on a certain restricted divisor problem for four binary linear forms.

Counting imaginary quadratic points via universal torsors

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin's conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A_3 with five lines given in P^4 by the equations vw - xy = vy + wy + xz = 0.

Inhomogeneous cubic congruences and rational points on del Pezzo surfaces

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.

Batyrev和Tschinkel例子中Manin猜想的几何例外集

The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example

针对Batyrev和Tschinkel提出的Fano 5-维Fermat三次曲面丛X，在特征0域上显式构造有限薄映射，证明其生成的薄集与Lehmann-Sengupta-Tanimoto猜想的Manin猜想例外集一致。

Evaluating Azumaya algebras on cubic surfaces

Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k) -> Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. We show that a suitably generic cubic surface over a number field, whose reduction at some prime is a cone, has no Brauer-Manin obstruction. This extends results of Colliot-Thélène, Kanevsky and Sansuc.

On the order three Brauer classes for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.

Universal torsors over Del Pezzo surfaces and rational points

We discuss Manin's conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method, we provide a proof of Manin's conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4. (Lectures at the summer school "Equidistribution in number theory", Montreal, August 2005)

Cubic surfaces failing the integral Hasse principle

Cubic surfaces failing the integral Hasse principle

研究仿射对角三次曲面的积分Brauer-Manin障碍，构造了该设定下积分Hasse原理的首个反例，并从三个自然角度计数反例分布及积分强逼近性质。

The density of rational points on Cayley's cubic surface

The Cayley cubic surface is given by the equation sum_{i=1}^4 X_i^{-1}=0. We show that the number of non-trivial primitive integer points of size at most B is of exact order B(log B)^6, as predicted by Manin's conjecture.

某些仿射三次曲面的Brauer群

Brauer groups of certain affine cubic surfaces

该研究针对特征0域上光滑三次曲面奇异超平面截面的补仿射曲面，确定了代数闭包上其Brauer群的Galois模结构，给出三线并截面对应的2、3阶超越元有理数域实例，并应用于整Hasse原理的整Brauer-Manin障碍研究。

On a certain non-split cubic surface

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.

虚二次域上 del Pezzo 曲面的整点

Integral Points on a del Pezzo Surface over Imaginary Quadratic Fields

本文研究虚二次域上 $\mathbf{A}_3$ 型四次 del Pezzo 曲面的整点，通过 universal torsors 进行高度有界计数并作几何解释，证明了相对奇点和直线的整点版 Manin 猜想类比。

对称仿射三次曲面上的整点

Integral points on symmetric affine cubic surfaces

本文证明对几乎所有整数$n$，由首一三次多项式定义的对称仿射三次曲面无整的Brauer-Manin障碍，从而支持Hasse原理。

三次曲面上点类的非结合Moufang环的一个例子

An example of a non-associative Moufang loop of point classes on a cubic surface

本文通过构造一个特定的三次曲面，证明了其点类模$(1-θ)^3$形成的交换Moufang环是非结合的，从而解决了Manin在50多年前提出的关于非结合Moufang环存在性的问题。

半整点在某一类奇异立方超曲面上的分布

The distribution of semi-integral points on a class of singular cubic hypersurfaces

本文研究了某一类奇异立方超曲面上半整点的计数函数渐进行为，并证明该渐进行为与Manin猜想在a-不变量和b-不变量上的预测一致。

Rational points and non-anticanonical height functions

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles.

On the number of Mordell-Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contain…

Generators for Cubic Surfaces with two Skew Lines over Finite Fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.

Zero-cycles and rational points on some surfaces over a global function field

Let F be a finite field of characteristic p. We consider smooth surfaces over F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4 variables with coefficients in F, with d prime to p. We prove : For such surfaces over F(t), the Brauer-Manin obstruction to the existence of a zero-cycle of degree one is the only obstruction. For d=3 (cubic surfaces), this leads to the same result for rational points. -- Soit F un corps fini de caractéristique p. Pour une surface lisse sur F(t) définie par une équation f+tg=0, où f et g sont deux formes de degré d sur F en 4 variables, ave…

奇异 del Pezzo 曲面上的整点

Integral points on singular del Pezzo surfaces

本文通过分类弱 del Pezzo 对，研究有界对数反典范高度整点，并在 $\mathbf{A}_1+\mathbf{A}_3$ 型四次奇异 del Pezzo 曲面上证明顾及奇点与直线的 Manin 型结论。

Del Pezzo surfaces over finite fields and their Frobenius traces

Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.

$R$-等价于立方面I：已有非平凡普遍等价的情况

$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

本文研究了具有非平凡普遍等价的立方面，发现2-进数立方面在特定条件下$R$-等价为平凡或指数2，并验证了特定例子的平凡性，同时探讨了生成AI在研究中的应用。