2021 Fiscal Year Research-status Report
p-adic Hodge theory, anabelian conjecture, and arithmetic Simpson correspondence
Project/Area Number |
19K03401
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Research Institution | Kyoto University |
Principal Investigator |
譚 福成 京都大学, 数理解析研究所, 講師 (00803587)
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Project Period (FY) |
2019-04-01 – 2024-03-31
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Keywords | p-adic Hodge theory |
Outline of Annual Research Achievements |
I have been working on the crystalline comparison theorem in p-adic Hodge theory, with non-trivial coefficients and in the relative setting, during the research period. I studied in depth the work of Faltings and Scholze on integral p-adic Hodge theory. Currently I am able to formulate integral results for etale and crystalline cohomology with non-trivial coefficients. The relative version of such results shall need further investigation. The current theory of prismatic cohomology of Bhatt and Scholze seems to be the right way of formulating an integral comparison theorem with non-trivial coefficients. In this direction, the coefficients will be the prismatic crystals, in terms of perfect complexes over Drinfeld’s stack, a generalization of the F-gauges of Fontaine-Jannsen.
It has become apparent to me that the language of higher categories, especially the non-abelian derived functors, will play an essential role in the study of p-adic Hodge theory, and more generally on arithmetic geometry and number theory. For my project, this provides the foundations to the nontrivial coefficients appearing on the crystalline comparison theorem, that is, the prismatic crystals, a mixed characteristic analogue of the classical F-crystals of Berthelot.Understanding of such objects require the knowledge on infinity-categories and E-infinity algebras, which I will study by organizing seminars with my collaborators.
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Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The research is going roughly as planned. The pandemic of covid19 has made it harder for me to meet with my collaborators and to attend the relevant workshops oversea.
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Strategy for Future Research Activity |
In the coming years, I will proceed according to my research plan as described in my research proposal. In the coming year, I will continue with providing the details of the proof of the current crystalline comparison theorem, as well as the works on prismatic crystals.
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Causes of Carryover |
Due the covid19 pandemic, my travel plans to the workshops oversea were canceled. Once the borders are reopened, which seems highly likely this year, I will be able to travel abroad and/or invite my collaborators from oversea to the joint research activities.
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