p-adic Abelian Stark conjectures at s=1
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 379-417.

A p-adic version of Stark’s Conjecture at s=1 is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our p-adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version is discussed in more detail and proved in two special cases.

Une version p-adique de la conjecture de Stark en s=1 est attribuée à J.-P. Serre et énoncée (de manière fautive) dans le livre de Tate sur cette conjecture. Dans le cas d’un corps de rayon réel sur un corps de nombres totalement réel, on présente ici une nouvelle conjecture de ce type, suivant plutôt la démarche de notre article précédent (et le travail de Rubin) sur la conjecture complexe abélienne. On étudie la cohérence de cette conjecture et on énonce des raffinements ‘sur ’, soit d’elle seule, soit en combinaison avec son analogue complexe. Enfin, la version ‘Weak Refined Combined’ fait l’objet d’une discussion plus détaillée et d’une démonstration dans deux cas particuliers.

DOI: 10.5802/aif.1891
Classification: 11R42,  11S40,  11R20,  11R27
Keywords: Stark conjecture, p-adic, L-function, zeta-function, abelian extension, unit, S-unit, regular, special value, totally real field
Solomon, David 1

1 King's College London, Department of Mathematics, Strand, London WC2R 2LS (Royaume-Uni)
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Solomon, David. $p$-adic Abelian Stark conjectures at $s=1$. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 379-417. doi : 10.5802/aif.1891. https://aif.centre-mersenne.org/articles/10.5802/aif.1891/

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