Géométrie algébrique, Théorie des nombres
Bounded Generation by semi-simple elements: quantitative results
[Engendrement borné par éléments semi-simples : résultats quantitatifs]
Comptes Rendus. Mathématique, Tome 360 (2022) no. G11, pp. 1249-1255.

Nous prouvons que pour un corps de nombres F, la distribution des points d’un ensemble Σ𝔸 F n admettant une paramétrisation purement exponentielle, par exemple un ensemble de matrices bornément engendré par des éléments semi-simples (diagonalisables), est de taille au plus logarithmique lorsqu’il est ordonné par hauteur. Par conséquent, on obtient qu’un groupe linéaire ΓGL n (K) sur un corps K de caractéristique zéro admet un paramétrisation purement exponentielle si et seulement s’il est de type fini et la composante connexe de sa clôture de Zariski est un tore. Nos résultats sont obtenus via une inégalité sur la hauteur des tuples minimaux d’un polynôme purement exponentiel. Un ingrédient clé de notre démonstration est une version forte par Evertse du théorème sur l’équation en S-unités.

We prove that for a number field F, the distribution of the points of a set Σ𝔸 F n with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group ΓGL n (K) over a field K of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal m-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse’s strengthening of the S-Unit Equation Theorem.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.376
Classification : 11D75
Corvaja, Pietro 1 ; Demeio, Julian L. 2 ; Rapinchuk, Andrei S. 3 ; Ren, Jinbo 4 ; Zannier, Umberto M. 5

1 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, via delle Scienze, 206, 33100 Udine, Italy
2 Departement Mathematik und Informatik, Universität Basel, 4051 Basel, Switzerland
3 Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
4 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
5 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
@article{CRMATH_2022__360_G11_1249_0,
     author = {Corvaja, Pietro and Demeio, Julian L. and Rapinchuk, Andrei S. and Ren, Jinbo and Zannier, Umberto M.},
     title = {Bounded {Generation} by semi-simple elements: quantitative results},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1249--1255},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G11},
     year = {2022},
     doi = {10.5802/crmath.376},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.376/}
}
TY  - JOUR
AU  - Corvaja, Pietro
AU  - Demeio, Julian L.
AU  - Rapinchuk, Andrei S.
AU  - Ren, Jinbo
AU  - Zannier, Umberto M.
TI  - Bounded Generation by semi-simple elements: quantitative results
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 1249
EP  - 1255
VL  - 360
IS  - G11
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.376/
DO  - 10.5802/crmath.376
LA  - en
ID  - CRMATH_2022__360_G11_1249_0
ER  - 
%0 Journal Article
%A Corvaja, Pietro
%A Demeio, Julian L.
%A Rapinchuk, Andrei S.
%A Ren, Jinbo
%A Zannier, Umberto M.
%T Bounded Generation by semi-simple elements: quantitative results
%J Comptes Rendus. Mathématique
%D 2022
%P 1249-1255
%V 360
%N G11
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.376/
%R 10.5802/crmath.376
%G en
%F CRMATH_2022__360_G11_1249_0
Corvaja, Pietro; Demeio, Julian L.; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M. Bounded Generation by semi-simple elements: quantitative results. Comptes Rendus. Mathématique, Tome 360 (2022) no. G11, pp. 1249-1255. doi : 10.5802/crmath.376. http://www.numdam.org/articles/10.5802/crmath.376/

[1] Amoroso, Francesco; Viada, Evelina Small points on subvarieties of a torus, Duke Math. J., Volume 150 (2009) no. 3, pp. 407-442 | DOI | MR | Zbl

[2] Bombieri, Enrico; Gubler, Walter Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006, xvi+652 pages | DOI | MR

[3] Corvaja, Pietro; Demeio, Julian L.; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M. Purely Exponential Parametrizations and their Group-theoretic Applications (2022) (in preparation)

[4] Corvaja, Pietro; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M. Non-virtually abelian anisotropic linear groups are not boundedly generated, Invent. Math., Volume 227 (2022) no. 1, pp. 1-26 | DOI | MR | Zbl

[5] Corvaja, Pietro; Zannier, Umberto M. Applications of Diophantine approximation to integral points and transcendence, Cambridge Tracts in Mathematics, 212, Cambridge University Press, 2018, x+198 pages | DOI | MR

[6] Duke, William; Rudnick, Zeev; Sarnak, Peter Density of integer points on affine homogeneous varieties, Duke Math. J., Volume 71 (1993) no. 1, pp. 143-179 | DOI | MR | Zbl

[7] Evertse, Jan-Hendrik; Győry, Kálmán Unit equations in Diophantine number theory, Cambridge Studies in Advanced Mathematics, 146, Cambridge University Press, 2015, xv+363 pages | DOI | MR

[8] Evertse, Jan-Hendrik; Schlickewei, Hans P.; Schmidt, Wolfgang M. Linear equations in variables which lie in a multiplicative group, Ann. Math., Volume 155 (2002) no. 3, pp. 807-836 | DOI | MR | Zbl

[9] Gorodnik, Alexander; Nevo, Amos Counting lattice points, J. Reine Angew. Math., Volume 663 (2012), pp. 127-176 | DOI | MR | Zbl

[10] Gorodnik, Alexander; Weiss, Barak Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal., Volume 17 (2007) no. 1, pp. 58-115 | DOI | MR | Zbl

[11] Hindry, Marc; Silverman, Joseph H. Diophantine geometry. An introduction, Graduate Texts in Mathematics, 201, Springer, 2000, xiv+558 pages | DOI | MR

[12] Maucourant, François Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices, Duke Math. J., Volume 136 (2007) no. 2, pp. 357-399 | DOI | MR | Zbl

[13] Prasad, Gopal; Rapinchuk, Andrei S. Existence of irreducible -regular elements in Zariski-dense subgroups, Math. Res. Lett., Volume 10 (2003) no. 1, pp. 21-32 | DOI | MR | Zbl

[14] Prasad, Gopal; Rapinchuk, Andrei S. Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces, Thin groups and superstrong approximation (Mathematical Sciences Research Institute Publications), Volume 61, Cambridge University Press, 2014, pp. 211-252 | MR | Zbl

[15] Prasad, Gopal; Rapinchuk, Andrei S. Generic elements of a Zariski-dense subgroup form an open subset, Trans. Mosc. Math. Soc., Volume 78 (2017), pp. 299-314 | DOI | MR | Zbl

[16] Rémond, Gaël Sur les sous-variétés des tores, Compos. Math., Volume 134 (2002) no. 3, pp. 337-366 | DOI | MR | Zbl

[17] Zannier, Umberto M. Lecture notes on Diophantine analysis, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 8, Edizioni della Normale, 2009, xvi+237 pages (With an appendix by Francesco Amoroso) | MR

Cité par Sources :