Title: Completed cohomology and Kato's Euler system for modular forms.

 

Abstract: Let f be a cuspidal Hecke eigenform of weight k ≥ 2, V_f the p-adic Galois representation attached to f. Using works of Colmez and Emerton on p-adic local Langlands and local-global compatibility, we can construct an element z_M (M stands for modular symbols) in the local Iwasawa cohomology of V_f^*. In this talk, I will show that the images of z_M under various dual exponential maps computes the classical L-values of the modular form f and its twists. We will then compare z_M with z_Kato - the image of Kato's Euler system in the local Iwasawa cohomology.