Relationships between p-unit constructions for real quadratic fields

الأصل · EN

Let K be a real quadratic field and let p be a prime number which is inert in K. Let Kₚ be the completion of K at p. In a previous paper, we constructed a p-adic invariant uC∈ Kₚ, and we proved a p-adic Kronecker limit formula relating uC to the first derivative at s=0 of a certain p-adic zeta function. By analogy with the p- adic Gross-Stark conjectures, we conjectured that uC is a p-unit in a suitable narrow ray class field of K. Recently, Dasgupta has proposed an exact p-adic formula for the Gross-Stark units of an arbitrary totally real number field. In our special setting, i.e., where one deals with a real quadratic number field, his construction produces a p-adic invariant uD∈ Kₚ. In this paper we show precise relationships between the p-adic invariants uC and uD. In order to do so, we extend Dasgupta's construction of uD to a broader setting.