Moduli Spaces and Rational Points

模空间和有理点

• 批准号: EP/K019279/1
• 负责人: Damiano Testa
• 金额: $ 12.07万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK019279%2F1
• 关键词: Moduli; Spaces; Rational; Points

One of the central questions in number theory is the solution of diophantine equation: to determine the set of all rational solutions of a system of polynomial equations with rational coefficients. The name diophantine is derived from name of Diophantus of Alexandria, of the third century AD, whose influential books "Arithmetica" shaped the development of number theory. From the point of view of algebraic geometry, the equations that Diophantus studied mostly define curves and his goal was to determine the set of integral (or rational) solutions.By virtue of many celebrated results, the case of rational points on curves is theoretically well-understood, and the motto "Geometry Determines Arithmetic" is fully justified. Indeed, an algebraic curve has a unique discrete invariant, its genus, taking non-negative integral values. If a curve has genus zero, then the question of determining whether it has rational points or not is completely algorithmic, and the set of all its rational points can be efficiently determined. If a curve has genus one, then the question of determining whether it has a rational point or not is typically feasible in concrete cases. There is a procedure to decide whether a curve of genus one has a point, but, if the curve does not have points, it is not known whether this procedure necessarily terminates. The finiteness of this procedure essentially relies on the finiteness of the Tate-Shafarevich groups. If a curve of genus one admits a point, then the set of its rational points can be endowed with a very natural structure of an abelian group. This group is finitely generated over number fields by the Mordell-Weil Theorem; explicit generators can again be found subject essentially to the Birch--Swinnerton-Dyer Conjecture. Finally, curves of genus at least two only have finitely many rational points, by Faltings' celebrated proof of the Mordell Conjecture.The situation is entirely different in higher dimensions. Bombieri and Lang formulated a conjecture implying that the distribution of rational points on varieties shares many similarities with the case of curves.Conjecture (Bombieri-Lang). The set of rational points of a smooth projective variety of general type over a number field is not Zariski dense.While this conjecture is very appealing, already in the case of surfaces, there is very little supporting evidence for it.The overall goal of this project is to study algebraic surfaces, mostly of general type, of special arithmetic interested, with the aim of gathering evidence for the Bombieri-Lang Conjecture. For this purpose we will compute the Picard groups and automorphism groups of various surfaces. We will use this information to look for curves of genus at most one on the surfaces, and determine the rational points on such curves. All this data will provide clues on possible modular interpretations of the surfaces: we will try to establish the modularity of these surfaces, trying first among moduli spaces of Abelian varieties and moduli spaces of vector bundles.

数论的核心问题之一是丢番图方程的解：确定具有有理系数的多项式方程组的所有有理解集。丢番图这个名字来源于公元三世纪亚历山大的丢番图斯，他有影响力的著作《算术》塑造了数论的发展。从代数几何的观点来看，丢番图研究的方程主要是定义曲线，他的目标是确定积分(或有理)解的集合。凭借许多著名的结果，曲线上的有理点的情况在理论上被很好地理解，并充分证明了座右铭“几何决定算术”。事实上，一条代数曲线有一个唯一的离散不变量，它的亏格取非负的整数值。如果一条曲线有亏格零，那么确定它是否有理性点的问题完全是算法问题，并且它的所有有理点的集合都可以有效地确定。如果一条曲线有亏格1，那么确定它是否有理性点的问题在具体情况下通常是可行的。有一个程序来确定亏格1的曲线是否有点，但如果曲线没有点，就不知道这个程序是否一定会终止。这一过程的有限性本质上依赖于Tate-Shafarevich群的有限性。如果亏格1的曲线允许一个点，则它的有理点集可以被赋予一个非常自然的阿贝尔群的结构。这个群是由Mordell-Weil定理在数域上有限地生成的；显式生成元本质上也可以根据Birch-Swinnerton-Dyer猜想找到。最后，根据Faltings著名的Mordell猜想证明，亏格至少两个的曲线只有有限多个有理点，在更高的维度上情况完全不同。Bombieri和Lang提出了一个猜想，即有理点在变元上的分布与曲线的情况有许多相似之处。数域上一般类型的光滑射影簇的有理点集不是Zariski稠密的。虽然这个猜想非常吸引人，但已经在曲面的情况下，很少有证据支持它。这个项目的总体目标是研究代数曲面，主要是一般类型的，具有特殊算术兴趣的，目的是为Bombieri-Lang猜想收集证据。为此，我们将计算各种曲面的Picard群和自同构群。我们将利用这些信息来寻找曲面上至多一条亏格的曲线，并确定这些曲线上的有理点。所有这些数据将为曲面的可能的模解释提供线索：我们将尝试建立这些曲面的模性，首先尝试在Abelian簇的模空间和向量丛的模空间之间进行尝试。

项目成果

Finite Weil restriction of curves

曲线的有限韦尔限制

• DOI: 10.1007/s00605-014-0711-6
• 发表时间: 2014
• 期刊: Monatshefte für Mathematik
• 作者: Flynn E

On Büchi's K3 surface

在 Büchi 的 K3 表面上

• DOI: 10.1007/s00209-014-1348-9
• 发表时间: 2014
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Artebani M

Reconstructing general plane quartics from their inflection lines

从拐点线重建一般平面四次方程

• DOI: 10.1090/tran/7599
• 发表时间: 2018
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Pacini M

Plane quartics with at least 8 hyperinflection points

具有至少 8 个超拐点的平面四次方程

• DOI: 10.1007/s00574-014-0077-3
• 发表时间: 2015
• 期刊: Bulletin of the Brazilian Mathematical Society, New Series
• 作者: Pacini M