Shintani zeta functions and gross-stark units for totally real fields
Let F be a totally real number field, and let p be a finite prime of F such that p splits completely in the finite abelian extension H of F. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a p-unit u in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to H/F. This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of u in Fpx/Ê, where FP denotes the completion of F at p and Ê denotes the topological closure of the group of totally positive units E of F. We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of u in Fpx.
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