Seminar
| Parent Program: | |
|---|---|
| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
ES Program Special Seminar: Eisenstein Cocycles For Imaginary Quadratic Fields
To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.
Let F be an imaginary quadratic field with integer ring O. We describe work in progress that aims to construct maps from the homology of Bianchi spaces for GL_2 over F to second K-groups of ray class fields of F and to show that they each factor through the quotient by the action of an Eisenstein ideal way from the level. This provides an analogue of maps varpi from the homology of modular curves to K_2 of cyclotomic fields (carrying Manin symbols to Steinberg symbols of cyclotomic units) and their conjectural Eisenstein property, proven under certain assumptions by Fukaya and Kato.
Our work grows out of a method Venkatesh and I developed to give a “motivic” construction of varpi (and certain zeta maps) that makes the Eisenstein property away from the level apparent. The basic idea is this: we construct Eisenstein cocycles on groups related to GL_2(O) valued in limits of second motivic cohomology groups of opens in products of two CM elliptic curves and then specialize these at torsion points in order to obtain the desired maps. Carrying this out, however, involves overcoming numerous intriguing and often substantial obstructions. This is joint work with E. Lecouturier, S. Shih, and J. Wang.
No Notes/Supplements Uploaded