The Darmon-Dasgupta units over genus fields and the Shimura correspondence

Abstract

We study a special case of the Gross-Stark conjecture (Gross. 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DO]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained. (C) 2010 Elsevier Inc. All rights reserved.