# On p-adic abelian Stark Conjectures at s=1

@article{Solomon2002OnPA,
title={On p-adic abelian Stark Conjectures at s=1},
author={David Solomon},
journal={Annales de l'Institut Fourier},
year={2002},
volume={52},
pages={379-417}
}
• D. Solomon
• Published 2002
• Mathematics
• Annales de l'Institut Fourier
Une version p-adique de la conjecture de Stark en s = 1 est attribuee a J.-P. Serre et enoncee (de maniere fautive) dans le livre de Tate sur cette conjecture. Dans le cas d'un corps de rayon reel sur un corps de nombres totalement reel, on presente ici une nouvelle conjecture de ce type, suivant plutot la demarche de notre article precedent (et le travail de Rubin) sur la conjecture complexe abelienne. On etudie la coherence de cette conjecture et on enonce des raffinements 'sur Z', soit d…

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## References

SHOWING 1-10 OF 35 REFERENCES
A Stark conjecture “over ${\bf Z}$” for abelian $L$-functions with multiple zeros
© Annales de l’institut Fourier, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions
Fonctions zeta p-adiques des corps de nombres abeliens réels
• Mathematics
• 1972
© Mémoires de la S. M. F., 1971, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l’accord
Twisted Zeta-Functions and Abelian Stark Conjectures
Abstract We present a new version “at s =1” of Rubin's refined, higher order Stark conjecture at s =0 for an abelian extension of number fields (K. Rubin, 1996, Ann. Inst. Fourier 46 , No. 1, 33–62).
Stark's Conjectures and Hilbert's Twelfth Problem
We give a constructive proof of a theorem of Tate, which states that (under Stark's Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real abelian
Cyclotomic Fields I and II
Contents: Character Sums.- Stickelberger Ideals and Bernoulli Distributions.- Complex Analytic Class Number Formulas.- The p-adic L-function.- Iwasawa Theory and Ideal Class Groups.- Kummer Theory
Galois Relations for Cyclotomic Numbers and p-Units
We study Galois relations between certain sets of cyclotomic numbers in real abelian fields. In one class of cases, a complete set of Galois generators for these relations is determined. These
The Shintani Cocycle: II. Partialζ-Functions, Cohomologous Cocycles andp-adic Interpolation☆☆☆
The “Shintani Cocyle”Φris further investigated under three headings. Firstly, a precise link with partialζ-values is given and non-triviality results are deduced for cocycles specializingΦr.
Numerical Verification of a p-Adic Abelian Stark Conjecture at s = 1
• preprint
• 2001