David Harbater, Daniel Krashen, Alena Pirutka
We investigate local-global principles for Galois cohomology, in the context of function fields of curves over semi-global fields. This extends work of Kato's on the case of function fields of curves over global fields.
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For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.
In response to a question of B. Poonen, we exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. In fact we can find such examples among Atkin-Lehner twists of certain elliptic modular curves and -- in positive characteristic -- Drinfeld modular curves. Our main tool is a refinement of the "Twist Anti-Hasse Principle" (TAHP). We also use TAHP to construct further Hasse Principle violations, for instance among curves over any number field of any given genus g which is at least 2.
In $2000$, Colliot-Thélène and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Thélène and himself. The main result in this paper generalizes the result of Colliot-Thélène and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to…
This is the final version, to appear in Commentarii Mathematici Helvetici.
The integral Hasse principle for stacky curves associated to a family of generalized Fermat equations
本文研究了关联于广义费马方程的堆叠曲线的积分哈斯原理，扩展了Darmon和Granville的结果，并发现族中正比例的曲线满足该原理。
We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $\mathbb{Z}$ that has an $\mathbb{R}$-point and a $\mathbb{Z}_p$-point for every prime $p$ but no $\mathbb{Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.
Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over \Q
我们展示了如何通过Atkin-Lehner对合构造有理数域上的Hasse原理的反例。一个特殊的例子是志村曲线X通过阿特金-莱纳对合w107与判别式23*107的不定有理四元数代数的商。关于这个对合的X被Q（sqrt{-23}）的二次扭曲也是Q上的哈塞原理的反例。
Baker and Rumely defined a notion of Arakelov-Green's functions on the Berkovich analytification of the projective line and established an Energy Minimization Principle. We extend their definition and show the Energy Minimization Principle for general smooth projective curves. As an application we get a generalization and a different proof of an equidistribution result by Baker and Petsche.
A generalized Mordell curve of degree $n \ge 3$ over $\bQ$ is the smooth projective model of the affine curve of the form $Az^2 = Bx^n + C$, where $A, B, C$ are nonzero integers. A generalized Fermat curve of signature $(n, n, n)$ with $n \ge 3$ over $\bQ$ is the smooth projective curve of the form $Ax^n + By^n + Cz^n = 0$ for some nonzero integers $A, B, C$. In this paper, we show that for each prime $p$ with $p \equiv 1 \pmod{8}$ and $p \equiv 2 \pmod{3}$, there exists a threefold $\cX_p \subseteq \bP^6$ such that certain rational points on $\cX_p$ produce infinite families of non-isomorphic…
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois repres…
There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k)=n and X has points over every completion of k.
An explicit algebraic family of genus-one curves violating the Hasse principle
证明了对Q中的任意t，P^2中的曲线5x^3 + 9y^3 + 10z^3 + 12（（t^12-t^4-1）/（t^12-t^8-1））^3（x+y+z）^3 = 0是亏格1曲线，违反了Hasse原理.给出了其Jacobian E_t的显式Weierstrass模型。每个E_t的Shafarevich-Tate群包含一个同构于Z/3 x Z/3的子群。
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。
On the Hasse principle for Shimura curves
设C是定义在数域K上的一条代数曲线，它具有正亏格且不含K-有理点.我们猜想存在某个扩域L，在该域上C违反Hasse原理，即，在本地到处都有点，但不是全局。本文证明了我们的猜想对除100个以外的所有形式为X^D_0（N）_{/Q}或X^D_1（N）_{/Q}的Shimura曲线都成立，其中D > 1，N是互素无平方的正整数.证明使用的变化定理弗雷，一个gonality界的阿布拉莫维奇，并分析当地的点的小程度。
We compare flat cohomology with crystalline syntomic complexes in two cases: 1) $p$-divisible groups over a separated $\mathbb F_p$-scheme with local finite $p$-bases, 2) semi-abelian schemes over a separated irreducible smooth curve.
We prove a non-archimedean analogue of the fact that a closed subvariety of a semi-abelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry.
An "Anti-Hasse Principle" for Prime Twists
给定一条局部处处有点的代数曲线C/Q，并赋予适当的对合，我们证明了C存在一个违反Hasse原理的素数二次扭曲的正密度族.这一结果特别适用于大多数模曲线X_0（N）的w_N-Atkin-Lehner扭数和某些Shimura曲线X^{D+}的w_p-Atkin-Lehner扭数。
We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $\p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $\p$.
A Hasse principle for $GL_2(\mathbb{F}_p)$ and Bloch's exact sequence for elliptic curves over number fields
本文研究数域上椭圆曲线的高阶Chow群SK_1(E)，通过分析范数映射核V(E)的模p结构，并结合模p Galois表示，给出了V(E)挠子群无限性的判定条件。
Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.
A family of curves over a discrete valuation ring is called semi-factorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the nodal case, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up centered at the non-regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend Raynaud's construction of the Néron (lft)-model of the jacobian of the generic fibre of a family of nodal curve…
We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational functions with their degrees. As a corollary, we show that the automorphism group of a tropical curve is isomorphic to the $\boldsymbol{T}$-algebra automorphism group of its rational function semifield, where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \}, \operatorname{max}, +)$ is the tropical semifield. Finally, we describe all semiring automorphi…
Cyclic p-groups and semi-stable reduction of curves in equal characteristic p>0
本文研究等特征$p>0$中曲线的$p$和$p^2$ -循环覆盖的半稳定约化.我们使用的主要工具是经典的Artin-Schreier-Witt理论的$p^n$ -循环覆盖的特征$p$。虽然本文的结果只涉及度$p$和$p^2$ -循环覆盖的情况下，我们开发的技术和框架，在一般的循环情况下，可以进行研究。
We prove that the Vologodsky integral of a mermorphic one-form on a curve over a $p$-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve.
In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution …
Local-global principle for over semiglobal fields
该研究比较半全局域上约化群下旋子的不同局部-全局原理，证明当群为收缩有理簇时，局部-全局原理对除子赋值完备化成立，丰富了代数数域相关理论。
We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central …
We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.
Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F.
We construct a semi-stable formal model of a wide open rigid curve with a semi-stable covering, and study the l-adic cohomology of the rigid curve. We describe the l-adic cohomology of the rigid curve using the l-adic cohomology of the irreducible components of a semi-stable reduction, and homology and cohomology of some graphs. We also prove the functoriality of the description for a finite flat morphism that is compatible with semi-stable coverings of wide open rigid curves.
We show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. In Theorem 1.1 we list integers can be attained as Milnor numbers of a given semi-quasi-homogeneous singularity.
Frey-Mazur theorem for semistable elliptic curves over function fields
该研究针对函数域上的非等平凡半稳定椭圆曲线，证明了Frey-Mazur型定理，即固定曲线亏格与域特征后，足够大的特征素数互素挠表示可确定其K-同源类。
Strongly Semistable Bundles on a Curve over a finite field
证明了有限域上光滑射影曲线上的主G丛是强半稳定的当且仅当它由曲线到G的基本群概型的表示定义。
We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves $φ: \varGamma \to \varGamma^{\prime}$, the rational function semifield of $\varGamma$ is finitely generated as a $φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$-algebra, where $φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of the rational…
We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent paper M. Kummer and K. Shaw defined the separating semigroup of $C$ as the set of all sequences $(d_1(f),\dots,d_n(f))$ where $f$ is a separating function and $d_i(f)$ is the degree of the restriction of $f$ to $A_i$. We describe the separating semigroup for hyperelliptic curves and for genus 3 curves.
Height and Intersection for a family of semi-stable curves
本文考虑一类半稳定曲线f：X \to B的算术Hodge指标定理，其中f_{\bf C}是光滑的，推广了算术曲面的Faltings-Hriljac的算术Hodge指标定理.
In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.
$\boldsymbol{T}$-algebra homomorphisms between rational function semifields of tropical curves
本文证明了两个热带曲线有理函数半域间的单射 $\boldsymbol{T}$-代数同态能诱导出这两个热带曲线间的满射态射。
Classification of semistable sheaves on a rational curve with one node
本文对单结点有理曲线上的（半）稳定层进行了分类。这些结果是基于Drozd和Greuel“Tame and wild射影曲线和向量丛的分类“中不可分解的无挠层的分类，其中层是用某些组合数据描述的.我们翻译的条件（半）稳定到这个组合语言和解决这样得到的问题。
We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.
Strong semistability of Higgs bundles over curves
本文研究曲线上的Higgs束强半稳定性，证明 genus ≤1 时半稳定Higgs束必强半稳定，而 genus ≥2 时构造了大秩非强半稳定的例子，补充了Lan-Sheng-Yang-Zuo和Langer的定理。
Semistability of Frobenius direct images over curves
设$X$是定义在特征为$p>0$的代数闭域$k$上的亏格为$g \geq 2$的光滑射影曲线.给定$X$上的一个半稳定向量丛$E$，我们证明了它在$X$的Frobenius映射$F$下的直接象$F\_*E$也是半稳定的.本文给出了稳定秩- $p$向量丛$F\_*L$的一个数值刻画，其中$L$是$X$上的线丛.
The Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves
构造了平滑曲线上半稳定向量丛的泛模空间的Hilbert紧化。希尔伯特紧化是一个格拉斯曼函数中的一个适当的希尔伯特曲线格式的某个开部分的GIT商。它拥有Teixidor要求的所有属性。
In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this form is equal to the number of positive divisors of $q+1$ and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so, we make use of two-point discrepancies and derive a criterion which applies to arbitrary curves over a finite field.
We study the existence of canonical Kähler metrics on the projectivisation of strictly Mumford semistable holomorphic vector bundles over a complex curve. We also provide an algebro-geometric characterization of these metrics.
Let $C$ be a smooth complex projective curve of genus $g \,\geq\, 2$ and $C_d$ its $d$-fold symmetric product. In this paper, we study the question of semi-orthogonal decomposition of the derived category of $C_d$. This entails investigations of the canonical system on $C_d$, in particular its base locus.
In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.
The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $μ(f)$ to $μ(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.
We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose the moduli space into strata which occur naturally as quotients modulo actions of certain algebraic groups. In some cases we give concrete geometric descriptions of the strata.
We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalises a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman--Morgan--Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.
Formal principle with convergence for rational curves of Goursat type
本文猜想括号生成有理曲线族的一般成员满足带收敛性形式原理，并在法丛最不正的 Goursat 型情形利用自然 ODE 与 Cartan 联络予以证明。
The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties
设f：C -> A是到半阿贝尔簇A的全纯曲线.则f（C）的Zebraki闭包是A的半阿贝尔子簇的平移（对数Bloch-Ochiai定理）。本文的目的是建立一个定量版本的上述结果，即，第二主要定理和亏损关系。
For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it is classically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, m K_C)$ is finitely generated in degree at most three. In this article, we study whether analogous statements hold for finite graphs and tropical curves. For any finite graph $G$, we show that the canonical semi-ring $R(G)$ is finitely generated but that the degree of generators are not bounded by a universal constant. For any hyperelliptic tropical curve $Γ$ with integer edge-length, we show that the canonical semi-ring $R(Γ)$ is not finitely ge…
Numerical bounds for semi-stable families of curves or of certain higher dimensional manifolds
给定射影曲线Y的开子集U和光滑曲线族f：V->U，在Y上具有半稳定约化，我们证明了对于秩>2的Hodge结构的子变差，Arakelov不等式必是严格的.对于家庭的n倍，我们证明了一个类似的结果的假设下，（n，0）组成部分的希格斯束定义纤维的双有理映射。
Holography principle and arithmetic of algebraic curves
根据全息原理（由于G.特胡夫特湖Susskind，J. Maldacena等人），在某些有边界流形上的量子引力弦论可以用边界上的共形场论来研究。只有少数数学上精确的结果证实了这个令人兴奋的计划。在本文中，我们解释从这个角度来看，最初出现在算术几何代数曲线的几个建设。我们证明了在算术无穷远处的双曲几何和Arakelov几何之间的关系包含了与黑洞的欧几里德AdS_3全息图完全相同的几何数据。此外，在欧氏AdS_2全息的情况下，我们给出了边界为非对易空间的体/边界对应的一些结果。
The minimal number of singular fibers of a semistable curves over P^1
本文证明了Beauville猜想：如果$f:S \to P^1$是亏格g>1的非平凡半稳定纤维化，则$f$至少有5个奇异纤维.我们还构造了一个亏格2的例子，其中有5个奇异纤维。这篇论文将发表在《代数几何杂志》上。
In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite field $F_q$ and P is a $F_q$-rational point of $X$. First we show how to evaluate the Feng-Rao Order Bound, which is a good estimation for the minimum distance of such codes. This bound is related to the classical Weierstrass semigroup of the curve $X$ at $P$. Further we focus our attention on the question to recognize the Weierstrass semigroups over fields of …
This paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. For Whitney's stratification in 1957, it partitions a real algebraic set into partial algebraic manifolds\cite{W}. In 1975 Hironaka reproved that a real algebraic set is triangulable and also generalized it to semi-algebraic sets, following the idea of Lojasiewicz's triangulation of semi-analytic sets in 1964. Following their examples and wondering how geometry looks like locally. this paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the f…
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