2022 Fiscal Year Research-status Report
p-adic Hodge theory, anabelian conjecture, and arithmetic Simpson correspondence
Project/Area Number |
19K03401
|
Research Institution | Kyoto University |
Principal Investigator |
上田 福大 京都大学, 数理解析研究所, 講師 (00803587)
|
Project Period (FY) |
2019-04-01 – 2024-03-31
|
Keywords | p-adic Hodge theory |
Outline of Annual Research Achievements |
The project is about 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. There has been some results in this direction, which generalizes my previous result published in 2019.
|
Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The research is slightly delayed, due to the rapid progress made on integral p-adic Hodge theory in the past several years. 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.
|
Strategy for Future Research Activity |
In the coming year, I will proceed according to my research plan as described in my research proposal.
More interestingly, I have discovered some connections between my previous research and K-theory, especially the Milnor K-theory for surfaces over number rings and S-integer rings. They have closer relationship to Galois cohomology of number fields, which has been my primary interest always. More concretely, these connections point to the direction of Schneider's conjecture on Galois cohomology of global Galois groups. Some cases were proven by Soule on the 1970's, known as Soule's vanishing theorem. Such a vanishing is the essential input in the work of M. Kim on the finiteness theorem of Siegel.
In the future I hope to focus on this direction leading to K-theory and Galois cohomology.
|
Causes of Carryover |
Due the covid19 pandemic, my travel plans to the workshops oversea were canceled. Since the borders are reopened this year, I will be able to travel abroad and/or invite my collaborators from oversea to the joint research activities.
|
Research Products
(1 results)