论文标题
表面束和截面猜想
Surface bundles and the section conjecture
论文作者
论文摘要
我们制定了Grothendieck截面的热带类似物：对于每个稳定的g> 2的稳定图G，每个场k，每个字段k，降低了g的通用曲线，都可以满足该部分的猜想。我们证明了许多猜想的情况。在这样做时，我们产生了许多曲线的示例，这些曲线满足了几何感兴趣领域的截面猜想，然后通过Chebotarev参数在P-Adic字段和数字字段上进行了。我们构建了两个GALOIS共同学类O_1和O_2，它们阻碍了PI_1段的存在以及理性点的存在。第一个是阿贝尔障碍物，与曲线的时期和曲线曲线曲线曲线的同一个学等级密切相关。第二个是2-尼尔替代性障碍物，似乎是新的。我们通过拓扑技术研究了这些类别的退化，并在这些类别阻碍切片的表面上产生了表面束的例子。然后，我们使用这些结构在P-ADIC字段和数字字段上产生曲线，其中每个类都阻碍了PI_1段，从而构成了合理点。在我们的几何结果中，有一个新的证据证明了g> 2属的通用曲线的截面猜想，也证明了对偶数属的通用曲线的截面猜想，其偶数曲线具有合理的学位一级分数（其中对截面的存在的障碍是真正的非阿贝利亚人）。
We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).
