Checking the p-adic stark conjecture when p is archimedean

Abstract

The p-adic Stark Conjecture makes good sense even when p is archimedean. When applied to the real subfield of a cyclotomic number field, it yields well-known information about the signs of cyclotomic units. Let k be a totally real cubic in which every unit is either totally positive or totally negative. In this case, the three conjugations σ1, σ2, and σ3 attached to the archimedean places of k are distinct in the narrow Hilbert Class Field H[Formula presented]over k. Let Ki be the fixed field of σi in H[Formula presented], and let εi be the Stark unit at the archimedean place p[Formula presented] that splits in Ki/k. We show that in this situation the local p[Formula presented]-Conjecture implies that εi is a square in Ki. We then numerically confirm this prediction by computationally verifying that εi is a square in a large number of examples; this is done by computing εi and it's conjugates over k to a large number of decimal places and then determining its irreducible polynomial over Q using a standard recognizer algorithm. The method used in computing the Lseries values at s = 0 attached to Ki/k is described in [DST].