Studies on Diophantine Geometry and Arakelov geometry
Project/Area Number 
17F17730

Research Category 
GrantinAid for JSPS Fellows

Allocation Type  Singleyear Grants 
Section  外国 
Research Field 
Algebra

Research Institution  Kyoto University 
Host Researcher 
森脇 淳 京都大学, 理学研究科, 教授 (70191062)

Foreign Research Fellow 
LIU CHUNHUI 京都大学, 理学(系)研究科(研究院), 外国人特別研究員

Project Period (FY) 
20171013 – 20200331

Project Status 
Granted (Fiscal Year 2019)

Budget Amount *help 
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2019: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2018: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2017: ¥600,000 (Direct Cost: ¥600,000)

Keywords  ディオファントス幾何 / アラケロフ幾何 / Diophantine Geometry / Arakelov Geometry 
Outline of Annual Research Achievements 
His collaboration work with some American scientists on the computer graphics and visualization has been accepted, where he uses his mathematical knowledge to accomplish the error analysis on an engineering problem. In May and June 2018, he visited France, Sweden and Germany. He discussed some subjects on the determinant method with Prof. Per Salberger at Chalmers University of Technology. After this visit, he told me that he has a better understand on the advantage and limitation of the determinant method in counting rational points problem, and Prof. Salberger has a better understand on the advantage of the intervention of Arakelov geometry. I suggested Chunhui Liu a recent work of Paul Vojta, " Roth's Theorem over arithmetic function fields". In this paper, Vojta generalized the classic Roth's theorem over finitely generated fields, where he uses an arithmetic height function introduced by me in 2000. He realized that it is hopeful to generalize some other results in Diophantine approximation over finitely generated fields, where we use the same basic setting. Before I introduced this work to him, he has studied some previous works on Diophantine approximation over higher dimensional arithmetic varieties, which will be useful to this program. In October, we followed the course of Prof. Huayi Chen on "Arakelov geometry over adelic curves" in Kyoto University, where Prof. Chen introduced his recent collaboration work with me. He understands this work very well, and this will be useful to the study on Diophantine approximation over finitely generated fields.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I am sure that all the work on counting rational points by the determinant method can be reformulated by the slope method in Arakelov geometry. But if we want to get an essentially better estimate, lots of work still need to be done. For example, we need to know for a projective geometrically integral scheme over a Dedekind ring, how many nonreduced reductions we have. Although in his Ph.D. thesis, he has given the optimal answer to this question for the case of reduced schemes, but the quantitative estimate is still not enough. In the program of Diophantine approximation over finitely generated fields, I expect he can work under the framework "Arakelov geometry over adelic curves" of Huayi Chen and me, which means he needs to rewrite at least parts of Vojta's work. Besides this, he need a better understand to more previous work on classic Diophantine approximation.

Strategy for Future Research Activity 
In order to have a understand on the rational points problem, he plans to take part in the activity " Thematic period: Reinventing rational points" in Paris from May to June. During this activity, he will meet lots of experts in rational points subjects, and they may provide my some inspiration. In the determinant method, a global control of multiplicity plays an important role, and a former work of O. Gabber, Q. Liu and D. Lorenzini may be useful for this. In order to refine one of his old works on counting multiplicities, he plans to visit Prof. Qing Liu at University of Bordeaux, and it is expected that Prof. Liu will provide him some suggestions. I will ask him to continue to study some other Diophantine work, in particular the Faltings Wustholz Theorem. When he has a proper enough understand to the classic theory, he will begin to build the Diophantine approximation theory over finitely generated fields or even adelic curves.

Report
(2 results)
Research Products
(6 results)