Duke Mathematical Journal

Shintani zeta functions and Gross-Stark units for totally real fields

Samit Dasgupta

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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 Fp×/E, where Fp denotes the completion of F at p and E 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 Fp×

Article information

Duke Math. J., Volume 143, Number 2 (2008), 225-279.

First available in Project Euclid: 26 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R37: Class field theory 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
Secondary: 11R80: Totally real fields [See also 12J15]


Dasgupta, Samit. Shintani zeta functions and Gross-Stark units for totally real fields. Duke Math. J. 143 (2008), no. 2, 225--279. doi:10.1215/00127094-2008-019. https://projecteuclid.org/euclid.dmj/1211819163

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