ANOTHER LOOK AT GROSS–STARK UNITS OVER THE NUMBER FIELD ℚ

Abstract

We provide another description of the Gross–Stark units over the rational field ℚ (studied in [B. Gross, p-Adic L-series at s = 0, J. Fac. Sci. Univ. Tokyo28(3) (1981) 979–994]) which is essentially a Gauss sum, using a p-adic multiplicative integral of the p-adic Kubota–Leopoldt distribution, and give a simplified proof of the Ferrero–Greenberg theorem (see [B. Ferrero and R. Greenberg, On the behavior of p-adic L-functions at s = 0, Invent. Math.50(1) (1978/79) 91–102]) for p-adic Hurwitz zeta functions. This is a precise analog for ℚ of Darmon–Dasgupta's work on elliptic units for real quadratic fields (see [H. Darmon and S. Dasgupta, Elliptic units for real quadratic fields, Ann. of Math. (2)163(1) (2006) 301–346]).