Shintani zeta-functions and Gross–Stark units for totally real fields
نویسنده
چکیده
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 . Stark has proposed a conjecture stating the existence of a p-unit 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 . Gross proposed a refinement of Stark’s conjecture which gives a conjectural formula for the image of Stark’s unit in F× p /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 propose a further refinement of Gross’ conjecture by proposing a conjectural formula for the exact value of Stark’s unit in F× p .
منابع مشابه
A Shintani-type Formula for Gross–stark Units over Function Fields
Let F be a totally real number field of degree n, and let H be a finite abelian extension of F . Let p denote a prime ideal of F that splits completely in H. Following Brumer and Stark, Tate conjectured the existence of a p-unit u in H whose p-adic absolute values are related in a precise way to the partial zeta-functions of the extension H/F . Gross later refined this conjecture by proposing a...
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