Checking the $\mathfrak{p}$-adic stark conjecture when $\mathfrak{p}$is archimedean

Abstract

The $\mathfrak{p}$-adic Stark Conjecture makes good sense even when $\mathfrak{p}$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 σ ₁, σ₂, and σ₃ attached to the archimedean places of k are distinct in the narrow Hilbert Class Field H ⁺ _k over k. Let K _i be the fixed field of σ_i in H ⁺ _k and let ε _i, be the Stark unit at the archimedean place $\mathfrak{p}_\infty ^{(\mathfrak{i})}$that splits in K _i k. We show that in this situation the local $\mathfrak{p}_\infty ^{(\mathfrak{i})}$-Conjecture implies that ε _i is a square in K _i . We then numerically confirm this prediction by computationally verifying that ε _i is a square in a large number of examples; this is done by computing ε _i 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 L-series values at s=0 attached to K _i k is described in [DST].

Partially supported by a grant from the National Security Agency.

Partially supported by a grant from the National Science Foundation.