Compositio Mathematica
(2003), no. 1, pp. 75-90.

The Local Stark Conjecture at a Real Place,
by Michael Reid

Compositio Mathematica
(2003), no. 1, pp. 75-90.

Abstract

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.

Reference

[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.
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Updated January 8, 2008.