[Les logarithmes plus et moins et la transformation d'Amice]
Nous donnons une nouvelle description des logarithmes p-adiques plus et moins définis par Pollack en termes de distributions. En particulier, si dénote la pré-image de sous la transformation d'Amice, nous donnons des formules explicites pour les valeurs pour tout et tout entier . Nos formules impliquent que la distribution correspond à une distribution étudiée par Koblitz en 1977. Par ailleurs, nous montrons qu'il existe une description similaire, due à Loeffler, pour des analogues à deux variables de ces logarithmes plus et moins.
We give a new description of Pollack's plus and minus p-adic logarithms in terms of distributions. In particular, if denote the pre-images of under the Amice transform, we give explicit formulae for the values for all and all integers . Our formulae imply that the distribution agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.
Accepté le :
Publié le :
@article{CRMATH_2017__355_9_942_0,
author = {Dion, C\'edric and Lei, Antonio},
title = {Plus and minus logarithms and {Amice} transform},
journal = {Comptes Rendus. Math\'ematique},
pages = {942--948},
publisher = {Elsevier},
volume = {355},
number = {9},
year = {2017},
doi = {10.1016/j.crma.2017.09.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.09.012/}
}
TY - JOUR AU - Dion, Cédric AU - Lei, Antonio TI - Plus and minus logarithms and Amice transform JO - Comptes Rendus. Mathématique PY - 2017 SP - 942 EP - 948 VL - 355 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.09.012/ DO - 10.1016/j.crma.2017.09.012 LA - en ID - CRMATH_2017__355_9_942_0 ER -
%0 Journal Article %A Dion, Cédric %A Lei, Antonio %T Plus and minus logarithms and Amice transform %J Comptes Rendus. Mathématique %D 2017 %P 942-948 %V 355 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.09.012/ %R 10.1016/j.crma.2017.09.012 %G en %F CRMATH_2017__355_9_942_0
Dion, Cédric; Lei, Antonio. Plus and minus logarithms and Amice transform. Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 942-948. doi : 10.1016/j.crma.2017.09.012. http://www.numdam.org/articles/10.1016/j.crma.2017.09.012/
[1] Distributions p-adiques associées aux séries de Hecke, Astérisque, Volume 1975 (1975) no. 24–25, pp. 119-131
[2] Λ-adic Gross–Zagier formula for supersingular primes, 2016 | arXiv
[3] The anticyclotomic main conjecture for elliptic curves at supersingular primes, J. Inst. Math. Jussieu, Volume 7 (2008) no. 2, pp. 291-325
[4] Iwasawa theory of elliptic curves at supersingular primes over -extensions of number fields, J. Reine Angew. Math., Volume 598 (2006), pp. 71-103
[5] Signed-Selmer groups over the -extension of an imaginary quadratic field, Can. J. Math., Volume 66 (2014) no. 4, pp. 826-843
[6] Two-variable p-adic L-functions of modular forms for non-ordinary primes, J. Number Theory, Volume 144 (2014), pp. 188-218
[7] Iwasawa theory for elliptic curves at supersingular primes, Invent. Math., Volume 152 (2003) no. 1, pp. 1-36
[8] p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York–Heidelberg, 1977
[9] p-Adic integration on ray class groups and non-ordinary p-adic L-functions, Iwasawa Theory 2012, Contrib. Math. Comput. Sci., vol. 7, Springer, Heidelberg, Germany, 2014, pp. 357-378
[10] Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes, 2015 (preprint) | arXiv
[11] On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J., Volume 118 (2003) no. 3, pp. 523-558
[12] Nonarchimedean measures associated with Dirichlet series, Mat. Sb. (N.S.), Volume 99(141) (1976) no. 2, pp. 248-260 (296, MR 0412114)
[13] Iwasawa main conjecture for supersingular elliptic curves, 2014 (preprint) | arXiv
Cité par Sources :
