Moduli Spaces and Rational Points

数论的中心问题之一是丢芬图方程的解：确定具有有理系数的多项式方程组的所有有理解的集合。丢番图这个名字来源于公元三世纪亚历山大的丢番图，他的著作《算术》影响了数论的发展。从代数几何的角度来看，丢芬图斯研究的方程大多是定义曲线的，他的目标是确定整（或有理）解的集合。由于许多著名的结果，曲线上有理点的情况在理论上得到了很好的理解，“几何决定算术”的格言是完全正确的。事实上，代数曲线有一个唯一的离散不变量，它的属，取非负的整数值。如果一条曲线的属为零，那么确定它是否有有理点的问题就完全是算法问题，它的所有有理点的集合就可以有效地确定。如果一条曲线的属为1，那么确定它是否有有理点的问题在具体情况下通常是可行的。有一个程序来决定一条属1的曲线是否有一个点，但是，如果曲线没有点，则不知道这个程序是否必然终止。这个过程的有限性本质上依赖于Tate-Shafarevich群的有限性。如果属1的曲线有一个点，那么它的有理点的集合可以被赋予一个非常自然的阿贝尔群结构。这个群是由莫德尔-韦尔定理在数域上有限生成的；显式生成器可以再次被发现基本上服从于Birch- Swinnerton-Dyer猜想。最后，由Faltings著名的莫德尔猜想证明，至少有两个属的曲线只有有限多个有理点。在高维中情况完全不同。Bombieri和Lang提出了一个猜想，暗示了有理点在变量上的分布与曲线的情况有许多相似之处。猜想(Bombieri-Lang)。数域上的一般光滑投影型有理点的集合不是Zariski稠密的。虽然这个猜想很有吸引力，但就表面而言，几乎没有证据支持它。这个项目的总体目标是研究代数曲面，主要是一般类型的，对特殊算术感兴趣的，目的是为Bombieri-Lang猜想收集证据。为此，我们将计算各种曲面的皮卡德群和自同构群。我们将使用这些信息来寻找曲面上最多为一个属的曲线，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，首先尝试在阿贝尔变体的模空间和向量束的模空间之间进行。

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.