Moduli Spaces and Rational Points

丢番图方程的解是数论中的一个中心问题：确定一个有理系数多项式方程组的所有有理解的集合。名称丢番图是来自亚历山大的丢番图的名字，公元第三世纪，其有影响力的书籍"算术"塑造了数论的发展。从代数几何的角度来看，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得到了很好的理解，"几何决定算术"的座右铭也得到了充分的证明。实际上，代数曲线有一个唯一的离散不变量，它的亏格，取非负整数值。若曲线亏格为零，则判定曲线是否有有理点的问题完全是算法问题，并且可以有效地判定曲线的所有有理点的集合。如果一条曲线有亏格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亏格为1的曲线是否有一个点，但是，如果曲线没有点，就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。如果亏格1的曲线允许一个点，那么它的有理点的集合可以被赋予一个阿贝尔群的非常自然的结构。这个群是由莫德尔-韦伊定理在数域上生成的;显式的生成元可以再次被发现，本质上服从伯奇-斯温纳顿-戴尔猜想。最后，由Faltings著名的Mordell猜想证明，亏格至少为2的曲线只有2个有理点，在高维中情况完全不同。Bubieri和Lang提出了一个猜想，暗示着簇上有理点的分布与曲线的情况有许多相似之处。一个数域上的一般类型的光滑射影簇的有理点集不是Zki稠密的。虽然这个猜想是非常吸引人的，但在曲面的情况下，支持它的证据很少。这个项目的总体目标是研究代数曲面，主要是一般类型的，特殊的算术感兴趣，目的是收集证据的Beri-Lang猜想。为此，我们将计算各种曲面的Picard群和自同构群。我们将使用这些信息来寻找曲面上亏格至多为一的曲线，并确定这些曲线上的有理点。所有这些数据将提供线索可能的模块化解释的表面：我们将试图建立这些表面的模块性，首先尝试之间的模空间的阿贝尔品种和模空间的向量丛。

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.