Kummer Covers with Many Points

与群概形相关的Kummer曲面

Kummer surfaces associated with group schemes

本研究在特征2情形下引入由群概形mu_2作用于有理尖点曲线自乘积得到的Kummer曲面，证明其恰为Artin不变量σ≤3的超奇异K3曲面，给出曲线构型刻画并说明其可作为单连通Enriques曲面的正规K3型覆盖。

Counting Rational Points on Kummer surfaces

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.

非特殊除子在具有任意分歧的Kummer覆盖上的构造用于LCP码

Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes

本文提出了一种新的框架，用于构造具有任意分歧的Kummer扩展中的非特殊除子，从而扩展了LCP AG码的构造方法，提高了其在密码学中的应用灵活性。

The Brauer group of Kummer surfaces and torsion of elliptic curves

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

Algebraic Geometric codes from Kummer Extensions

For Kummer extensions defined by $y^m = f (x)$, where $f (x)$ is a separable polynomial over the finite field $\mathbb{F}_q$, we compute the number of Weierstrass gaps at two totally ramified places. For many totally ramified places we give a criterion to find pure gaps at these points and present families of pure gaps. We then apply our results to construct many points algebraic geometric codes with good parameters.

Zariski K3 surfaces

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $α_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.

Kummer surfaces and the computation of the Picard group

We test R. van Luijk's method for computing the Picard group of a $K3$ surface. The examples considered are the resolutions of Kummer quartics in $\bP^3$. Using the theory of abelian varieties, in this case, the Picard group may be computed directly. Our experiments show that the upper bounds provided by R. van Luijk's method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces $V$ of Picard rank 18, we have $\smash{\rk\Pic(V_{\bar\bbF_{p}}) \geq 20}$ for a set of primes of …

Nodal quintic surfaces and lines on cubic fourfolds

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville, Murre, and Voisin. Results due to Izadi and Shen are particularly close to some of the statements. In t…

数域上最大Kummer扩张中概形的上同调

Cohomology of varieties over the maximal Kummer extension of a number field

本文研究了数域上最大Kummer扩张中概形的上同调性质，证明了其有限不变类及阿贝尔簇 torsion 的有限性，同时指出非阿贝尔可解扩张的torsion有限性不受伽罗瓦群控制。

代数曲线的几何阿代尔上的 Kummer 理论

Kummer theory over the geometric adeles of an algebraic curve

本文利用几何阿代尔环的 Kummer 理论，通过证明 Picard 群平凡性、建立局部-全局原理及寻找分类不变量，对代数闭域上代数曲线的有限阿贝尔 Galois 覆叠进行了纯代数刻画。

论线与伪线构形及球商

On line and pseudoline configurations and ball-quotients

本文证明射影平面中几乎不存在使高阶 Kummer 覆盖为球商的实直线构形，并利用 Shnurnikov 不等式给出拓扑 $(n_k)$-构形在 $n<27$ 和 $n<41$ 时的不存在性结果。

几何Kummer $K3$曲面上零环的局部和局部到全局原理

Local and local-to-global Principles for zero-cycles on geometrically Kummer $K3$ surfaces

研究$p$进域上几何Kummer $K3$曲面的零环Chow群结构，证明其可分解为可除群与有限群的直和，并给出局部到全局原理的首次无条件证据。

Birational geometry of blow-ups of projective spaces along points and lines

Consider the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism $ϕ_X$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $ϕ_X$ on $S$ realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, w…

Galois modules, ideal class groups and cubic structures

We establish a connection between the theory of cyclotomic ideal class groups and the theory of "geometric" Galois modules and obtain results on the Galois module structure of coherent cohomology groups of Galois covers of varieties over Z. In particular, we show that an invariant that measures the obstruction to the existence of a virtual normal integral basis for the coherent cohomology of such covers is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a normal integral basis is closely connected to the truth of …

On the number of rational points on curves over finite fields with many automorphisms

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.

论Skabelund的Suzuki曲线与Ree曲线的射线类域覆盖

On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves

本文证明了Skabelund构造的Suzuki与Ree曲线的射线类域覆盖分别等于其循环覆盖，对所有可容许的q成立，结合Kummer标准形式与Galois群中心性等工具。

Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for 0<m<n follows from the non-existence of (non-zero) fixed points in certain representations of A_{n+1}. For n<4 we provide an explicit crepant resolution X in characteristics different from 2,3. The key point is that Y can be realized as a double cover of P^n branched along a hypersurface of degr…

Wild monodromy and automorphisms of curves

Let $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with field of fractions $K$ containing the $p$-th roots of unity. This paper is concerned with semi-stable models of $p$-cyclic covers of the projective line $C \la \PK$. We start by providing a new construction of a semi-stable model of $C$ in the case of an equidistant branch locus. If the cover is given by the Kummer equation $Z^p=f(X_0)$ we define what we called the monodromy polynomial ${\mathcal L}(Y)$ of $f(X_0)$; a polynomial with coefficients in $K$. Its zeros are key to obtaining a semi-stable model of $C$.…

Computing in Jacobians of projective curves over finite fields

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Pic…

Coverings of curves with asymptotically many rational points

The number A(q) is the upper limit of the ratio of the maximum number of points of a curve defined over $\Fq$ to the genus. By constructing class field towers with good parameters we present improvements of lower bounds of A(q) for q an odd power of a prime.

An elementary abelian $p$-cover of the Hermitian curve with many automorphisms

The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.

相对完备化的幂单Chabauty-Kim-Kantor方法

The Unipotent Chabauty-Kim-Kantor Method for Relative Completions

该研究基于Kantor的相对完备化思路，开发了新的p进解析方法，验证了相关维数不等式，得到Faltings与Siegel定理的新条件证明，还取得多项关键技术贡献。

The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…