L-functions and other arithmetic invariants of curves of genus greater than or equal to 3

本课题的主题是利用曲线的半稳定约简技术来研究奇异点的解算，特别是二维野商奇异点的解算。一个中心的例子来自于对p进域上光滑投影曲线的研究。在这里，我们通过商构造来寻求将曲线的合适的半稳定模型与正则模型联系起来。这种构造产生了商奇点，并提出了求解商奇点的新方法。更一般地说，目标是将这种构造扩展到具有相同特征p的曲面上，并扩展到更高的维度。我们计划利用这项研究的结果来开发实用的算法来计算曲线的半稳定模型。我们还希望更好地了解我们工作中出现的商奇点的结构，从而能够在迄今为止无法使用蛮力计算的情况下，通过算法计算它们的分辨率。

The main theme of the proposed project is to use techniques related to semistable reduction of curves to study resolution of singularities, in particular wild quotient singularities in dimension two. A central case arises from the study of smooth projective curves over a p-adic field. Here we search to relate a suitable semistable model of the curve to a regular model via a quotient construction. This construction gives rise to quotient singularities, and we propose new methods for resolving them. More generally, the goal is to extend this construction also to surfaces in equal characteristic p, and to higher dimension. We plan to use the results of this study to develop practical algorithms to compute semistable models of curves. We also expect to obtain a better insight into the structure of the quotient singularities arising in our work, and thus be able to algorithmically compute their resolution, in cases that were so far inaccessible to brute force calculation.