The Local Stark Conjecture at a Real Place

The Local Stark Conjecture at a Real Place, by Michael Reid
Compositio Mathematica 137 (2003), no. 1, pp. 75-90.
[DOI] [Math Reviews] [Zentralblatt]
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Dummit and Hayes [1] noticed empirically that, in a number of computational examples of Stark's conjecture, the Stark unit is a square. Subsequently, they showed how this would follow from the truth of the p-adic Stark conjecture. In this paper, we extend Dummit and Hayes' result to a much more general situation. Let k be a totally real number field, L an abelian extension in which the real place v ramifies, and K the fixed field of the corresponding complex conjugation. If the subgroup of Gal(L/k) generated by all complex conjugations has rank strictly less than [k : Q] (and some other very minor conditions hold), then the p-adic Stark conjecture (for p = v) implies that the Stark unit for K/k is a square in K . The significance, as noted by Dummit and Hayes, is that the abelian part of Stark's conjecture holds automatically in this case.
[1] David S. Dummit, David R. Hayes, Checking the p-adic Stark conjecture when p is Archimedean, Algorithmic number theory (Talence, 1996), pp. 91-97, Lecture Notes in Computer Science, vol. 1122. [Math Reviews] [Zentralblatt]
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Updated January 8, 2008.