Edged crystalline cohomology
Final Report Abstract
This report summarizes my research work funded by the Walter Benjamin Fellowship under the ´ ´ project titled Edged Crystalline Cohomology conducted in the equipe de Theorie des Nombres at the IMJ-PRG, Jussieu, Paris, from September 2021 to August 2023. The project was supervised by Matthew Morrow and focused on the interaction between crystalline, rigid, and flat cohomology and its applications to Brauer groups, abelian varieties, and Shimura varieties in positive characteristic. The primary goals of the project have been the followings. • To advance the theory of crystalline cohomology for open varieties through the framework of edged crystalline cohomology. • To investigate the boundedness of the p-primary torsion in the Brauer group of abelian varieties, following questions posed by Skorobogatov and Zarhin and developing a variant of the Tate conjecture for flat cohomology. • To solve Chai–Oort Hecke orbit conjecture together with Pol van Hoften, contributing to the understanding of Newton polygon stratification and the behavior of Hecke orbits in the reduction modulo p of Shimura varieties. For this purpose we used the parabolicity conjecture, that we previously solved in 2020.
