Wach modules over Iwasawa algebras and multivariable (φ,Γ)-modules

• Applicant: Dr. Rustam Steingart
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 536703837

In number theory one studies solutions to polynomial equations over the integers. The symmetries between different solutions play an important role and are described by Galois groups. Instead of studying these groups directly, it is often more reasonable to study how these groups act on other objects such as vector spaces. In this case one speaks of Galois representations. In some cases these Galois representations can be described by so-called (phi,gamma)-modules. From the (phi,gamma)-modules one can reconstruct many important invariants such as Iwasawa cohomology. We put a special emphasis on (phi,gamma)-modules over the character variety of Schneider and Teitelbaum attached to crystalline representations. The goal of this research is the description of the crystalline representations among (phi,gamma)-modules over the character variety in terms of suitable structures over the Iwasawa algebra. Subsequently we plan to study Iwasawa cohomology and compare it to its analytic analogue. The additional structures can contribute to a proof of the Euler-Poincaré-formula for analytic cohomology. At last, we will connect the objects to the theory of multivariable (phi,gamma)-modules and aim to compare the theories of Berger and Zábrádi.

• DFG Programme: WBP Fellowship
• International Connection: France
• Host: Professor Dr. Laurent Berger