Soit un nombre premier. Soit tel que , soient des caractères de conducteur premier à ; notons le caractère de Teichmüller. Pour tout entre et et pour tout entre et , on pose
Soit et soit un premier de l’anneau de valuation de . Pour tout notons la série d’Iwasawa associée à et sa réduction modulo . Finalement soit une clôture algébrique de . Nous montrons que si les caractères sont distincts modulo , alors et les séries , sont linéairement indépendantes sur un certain corps qui contient .
Let be a prime. Let such that , let be characters of conductor not divided by and let be the Teichmüller character. For all between and , for all between and , set
Let and let be a prime of the valuation ring of . For all let be the Iwasawa series associated to and its reduction modulo . Finally let be an algebraic closure of . Our main result is that if the characters are all distinct modulo , then and the series are linearly independent over a certain field that contains .
@article{AIF_2010__60_5_1831_0, author = {Angl\`es, Bruno and Ranieri, Gabriele}, title = {On the linear independence of $p$-adic $L$-functions modulo $p$}, journal = {Annales de l'Institut Fourier}, volume = {60}, year = {2010}, pages = {1831-1855}, doi = {10.5802/aif.2573}, zbl = {1219.11162}, mrnumber = {2766231}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2010__60_5_1831_0} }
Anglès, Bruno; Ranieri, Gabriele. On the linear independence of $p$-adic $L$-functions modulo $p$. Annales de l'Institut Fourier, Tome 60 (2010) pp. 1831-1855. doi : 10.5802/aif.2573. http://gdmltest.u-ga.fr/item/AIF_2010__60_5_1831_0/
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