发表机构
IBM Deutschland Research & Development; Concordia University; Google DeepMind（IBM德国研究与开发; 康考迪亚大学; 谷歌DeepMind）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了整数环上算术曲面的Golod--Shafarevich塔，得到带截面且几何étale基本群无限的曲面，并取一般纤维获得有理数域上曲线，其有理点完全分裂且亏格趋于无穷，对数导子线性增长，回答了多个开放问题。
AI 中文摘要
我们在整数环上构造了算术曲面的Golod--Shafarevich塔。特别地，我们得到了一个带截面的算术曲面，其几何étale基本群为无限，回答了Bost和Charles提出的关于此类曲面存在性的问题。取一般纤维得到有理数域上的曲线，这些曲线具有无限的几何étale塔，且有理点在其中完全分裂。这回答了Ihara以及Frey、Kani和Völklein提出的问题。取塔的一般纤维，我们还得到有理数域上的曲线序列，其亏格趋于无穷，而对数导子（logarithmic conductors）的增长不超过亏格的线性函数，回答了Venkatesh在2022年PCMI提出的问题。作者们自2022年起共同研究此问题，并发展了一个总体策略。他们仅在与GPT5.6 Sol、Fable和Aletheia（一个由Google DeepMind开发的、基于Gemini的内部智能体）合作后才得以实现。
英文摘要
We construct Golod--Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamental group, answering a question raised by Bost and Charles about the existence of such surfaces. Taking generic fibres gives curves over the rationals with infinite geometric etale towers in which a rational point splits completely. This answers questions posed by Ihara and by Frey, Kani and Völklein. Taking the generic fibres of the tower, we also obtain sequence of curves over the rationals whose genera goes to infinity and whose logarithmic conductors grows no more than linearly in the genera, answering a question asked by Venkatesh at PCMI in 2022. The authors worked together on this problem since 2022 and developed a general strategy. They were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind.
