2017 Fiscal Year Annual Research Report
Studies on Diophantine Geometry and Arakelov geometry
| Project/Area Number |
17F17730
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| Research Institution | Kyoto University |
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Principal Investigator |
森脇 淳 京都大学, 理学研究科, 教授 (70191062)
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| Co-Investigator(Kenkyū-buntansha) |
LIU CHUNHUI 京都大学, 理学(系)研究科(研究院), 外国人特別研究員
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| Project Period (FY) |
2017-10-13 – 2020-03-31
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| Keywords | Diophantine Geometry / Arakelov Geometry |
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| Outline of Annual Research Achievements |
Counting rational points of bounded height in arithmetic varieties is a central topic in quantitative arithmetic, and the determinant method is one of the key ingredients of studying it. In order to study it by this method, it is very important to understand the complexity of the singular locus of arithmetic varieties. We have the following observation: a projective variety cannot contain too many singular points with large multiplicities, in other words, the singular locus cannot be too complicated. Since the consideration of multiplicities plays an important role in the determinant method, we should give a global numerical description of it. We transform the central issue to some counting multiplicities problems.
Next, it is an ingredient to choose the counting function. In order to detect the singular locus, we should require the counting function f satisfying f(1)=0. We choose a counting function having the polynomial asymptotic, and we require the asymptotic polynomial has the degree as large as possible. At the same time, we also require the upper bound has an as small as possible dependence on the degree of the original varieties.
In Liu's preprint "comptage des multiplicites dans une hypersurface sur un corps fini", he introduces a notion called "intersection tree" to give a numerical description of this observation, and in this paper the case over finite fields is solved. During Oct 2017- March 2018, by the similar method, he solves the case over number fields (considering rational points with bounded height) with the collaboration with Dr. Hao WEN.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
In the historical literature, counting rational points problems are widely considered, and there are many excellent results in this subject and still many important open questions. However, if we generalize the counting function (for counting rational points problem, the counting function is constant function), there are not so many results. In addition, it is another question whether these kinds of generalization is important or useful. In our counting multiplicities problem, we choose a counting function f with polynomial asymptotic and f(1)=0, hence it is possible to detect the complexity of singular locus by it. This method gives a new viewpoint to understand the complexity of singular locus, and it will be useful in considering the density of rational points of bounded height of arithmetic varieties. In Liu's previous work, "comptage des multiplicites dans une hypersurface sur un corps fini" (arxiv: 1606.09337), he gave an almost optimal response to the finite fields case. During his post-doc research in Kyoto, he gave an almost optimal response to the number field case, which is an original result.
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| Strategy for Future Research Activity |
We have already understood the contribution of singular locus to the counting rational points problem, which means that maybe the machine of the determinant method becomes stronger, and we may get more fruitful results in this area. In the following financial year, Liu will focus on some conjectures of R. Heath-Brown about the density of rational points in arithmetic varieties. If we reformulate the discriminant method by the slope method of Arakelov geometry, it is easier to work over number fields and get some explicit estimates. So the reformulated discriminant method will be useful in some quantitative problems where we care the dependence of some geometric invariants of arithmetic varieties, and even it is hopeful to consider the problems of counting algebraic points or cases over rational functions fields. In this financial year, Liu will try to consider these subjects.
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