We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
Nous demontrons que le dixième problème de Hilbert pour un anneau d’entiers dans un corps de nombres admet une réponse négative si satisfait à deux conditions arithmétiques (existence d’un ensemble dit division-ample et d’une courbe elliptique de rang un sur ). Nous lions les ensembles division-ample à l’arithmétique des variétés abéliennes.
@article{JTNB_2005__17_3_727_0,
author = {Gunther Cornelissen and Thanases Pheidas and Karim Zahidi},
title = {Division-ample sets and the {Diophantine} problem for rings of integers},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {727--735},
publisher = {Universit\'e Bordeaux 1},
volume = {17},
number = {3},
year = {2005},
doi = {10.5802/jtnb.516},
mrnumber = {2212121},
zbl = {05016583},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/}
}
TY - JOUR AU - Gunther Cornelissen AU - Thanases Pheidas AU - Karim Zahidi TI - Division-ample sets and the Diophantine problem for rings of integers JO - Journal de théorie des nombres de Bordeaux PY - 2005 DA - 2005/// SP - 727 EP - 735 VL - 17 IS - 3 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/ UR - https://www.ams.org/mathscinet-getitem?mr=2212121 UR - https://zbmath.org/?q=an%3A05016583 UR - https://doi.org/10.5802/jtnb.516 DO - 10.5802/jtnb.516 LA - en ID - JTNB_2005__17_3_727_0 ER -
%0 Journal Article %A Gunther Cornelissen %A Thanases Pheidas %A Karim Zahidi %T Division-ample sets and the Diophantine problem for rings of integers %J Journal de théorie des nombres de Bordeaux %D 2005 %P 727-735 %V 17 %N 3 %I Université Bordeaux 1 %U https://doi.org/10.5802/jtnb.516 %R 10.5802/jtnb.516 %G en %F JTNB_2005__17_3_727_0
Gunther Cornelissen; Thanases Pheidas; Karim Zahidi. Division-ample sets and the Diophantine problem for rings of integers. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 727-735. doi : 10.5802/jtnb.516. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/
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