Speaker: Jeehoon Park (BU)
Title: The rationality of elliptic units for real quadratic fields over genus fields
Abstract: Darmon and Dasgupta constructed elliptic units for real quadratic fields, which is an analogue of elliptic units for imaginary quadratic fields in the real quadratic fields and conjecturally yields the explicit class field theory for real quadratic fields, using p-adic analytic methods. Since the construction itself is p-adic, we don't know apriori that the elliptic units for real quadratic fields are global p-units. But they conjectured that they are actually global p-units in the ring class field of real quadratic fields. In this talk, I will explain how to prove some special case of this conjecture, namely some linear combinations of elliptic units are global over genus fields of real quadratic fields. First part of my talk, I will generalize the Darmon and Dasgupta's definition to define elliptic units attached to the modular discriminant using Borel-Serre completion of the upper plane and connection of elliptic units to p-adic family of half-integral weight Eisenstein series. Second part of the talk, I will explain the key ideas of the proof. The key ingredients of the proof are the p-adic version of Kronecker limit formula, some factorization formula of Dedekind zeta function of real quadratic field, and some proven case of Gross-Stark conjecture by Gross.