Fonctions L p -adiques et irrationalité
Bel, Pierre1
1 Institut de Mathématiques de Bordeaux, UMR 5251, Université Bordeaux 1, 351, cours de la Libération, 33405 Talence cedex, France
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 189-227.

We give a minoration of the dimension of the vector space spanned on a cyclotomic field by the values of p-adic Hurwitz zeta function. As a corollary, we obtain the existence of irrationality values of p-adic L functions. The proof uses hypergeometric series and a criterion of linear independence.

Classification : 11J72, 11J61
Bel, Pierre 1

1 Institut de Mathématiques de Bordeaux, UMR 5251, Université Bordeaux 1, 351, cours de la Libération, 33405 Talence cedex, France 
@article{ASNSP_2010_5_9_1_189_0,
     author = {Bel, Pierre},
     title = {Fonctions $L$ $ $p$-adiques et irrationalit\'e},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {189--227},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {1},
     year = {2010},
     mrnumber = {2668878},
     zbl = {1203.11051},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2010_5_9_1_189_0/}
}
TY  - JOUR
AU  - Bel, Pierre
TI  - Fonctions $L$ $ $p$-adiques et irrationalité
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 189
EP  - 227
VL  - 9
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_1_189_0/
LA  - en
ID  - ASNSP_2010_5_9_1_189_0
ER  - 
%0 Journal Article
%A Bel, Pierre
%T Fonctions $L$ $ $p$-adiques et irrationalité
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 189-227
%V 9
%N 1
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_1_189_0/
%G en
%F ASNSP_2010_5_9_1_189_0
Bel, Pierre. Fonctions $L$ $ $p$-adiques et irrationalité. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 189-227. http://archive.numdam.org/item/ASNSP_2010_5_9_1_189_0/

[1] K. Ball and T. Rivoal, Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146 (2001), 193–207. | MR

[2] F. Beukers, Irrationality of some p-adic L-values, Acta Math. Sin. (Engl. Ser.) 24 (2009), 663–686. | MR | Zbl

[3] F. Calegari, Irrationality of certain p-adic periods for small p, Int. Math. Res. Not. 20 (2005), 1235–1249. | MR | Zbl

[4] H. Cohen, “Number Theory”, Vol. II, Analytic and Modern Tools, Graduate Texts in Mathematics, 240, Springer, New-York, 2007. | MR

[5] N. I. Feldman and Yu. V. Nesterenko, “Transcendental Numbers”, In: Encyclopaedia of Mathematical Sciences, Vol. 44, Springer, New-York, 1998. | MR

[6] R. Marcovecchio, Linear independence of forms in polylogarithms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006), 1–11. | EuDML | Numdam | MR | Zbl

[7] Y. V. Nesterenko, Linear independence of numbers, Mosc. Univ. Math. Bull. 40 (1985), 69–74; traduction Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1 (1985), 46–54. | MR | Zbl

[8] T. Rivoal, Simultaneous polynomial approximations of the Lerch function, Canadian J. Math. 61 (2009), 1341–1356. | MR | Zbl