Time: October 16, Wednesday, 16:00-17:00

Venue: Lecture Hall

Title: Eisenstein cocycles for imaginary quadratic fields

Speaker: Emmanuel Lecouturier (Westlake University)

Abstract: Sharifi constructed a beautiful explicit map $\varpi_N$ from the first singular homology group of the modular curve $X_1(N)$ to the second algebraic K-group of the cyclotomic field $\Q(\zeta_N)$. He conjectured that this map is annihilated by a certain Eisenstein ideal, which was proved recently by Sharifi and Venkatesh using the motivic cohomology of the square of the torus. In this talk, we explain a construction of an analogue of $\varpi_N$ in the setting of Bianchi 3-folds attached to imaginary quadratic fields. This map is not as explicit as in the case of modular curves due to the lack of a canonical and explicit presentation of the first homology group of Bianchi 3-folds in general. Our construction, and the proof of the Eisenstein property, relies on techniques similar to the ones of Sharifi and Venkatesh but replacing the square of a torus by products of CM elliptic curves. This is a joint work in progress with Romyar Sharifi, Sheng-Chi Shih, and Jun Wang.