Rational fixed points for linear group actions

Pietro Corvaja

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 561-597
  • ISSN: 0391-173X

Abstract

top
We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group G , an affine variety V and a finite map π : V → G , all defined over a finitely generated field κ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set π ( V ( κ ) ) contains a Zariski dense sub-semigroup Γ ⊂ G ( κ ) ; namely, there must exist an unramified covering p : G ˜ → G and a map θ : G ˜ → V such that π ∘ θ = p . In the case κ = ℚ , G = 𝔾 a is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of GL N admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.

How to cite

top

Corvaja, Pietro. "Rational fixed points for linear group actions." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 561-597. <http://eudml.org/doc/272251>.

@article{Corvaja2007,
abstract = {We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\rightarrow G$, all defined over a finitely generated field $\kappa $ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi (V(\kappa ))$ contains a Zariski dense sub-semigroup $\Gamma \subset G(\kappa )$; namely, there must exist an unramified covering $p:\tilde\{G\}\rightarrow G$ and a map $\theta :\tilde\{G\}\rightarrow V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb \{Q\}$, $G=\mathbb \{G\}_\{a\}$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of $\operatorname\{GL\}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.},
author = {Corvaja, Pietro},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Algebraic groups; Hilbert irreducibility},
language = {eng},
number = {4},
pages = {561-597},
publisher = {Scuola Normale Superiore, Pisa},
title = {Rational fixed points for linear group actions},
url = {http://eudml.org/doc/272251},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Corvaja, Pietro
TI - Rational fixed points for linear group actions
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 561
EP - 597
AB - We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\rightarrow G$, all defined over a finitely generated field $\kappa $ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi (V(\kappa ))$ contains a Zariski dense sub-semigroup $\Gamma \subset G(\kappa )$; namely, there must exist an unramified covering $p:\tilde{G}\rightarrow G$ and a map $\theta :\tilde{G}\rightarrow V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb {Q}$, $G=\mathbb {G}_{a}$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of $\operatorname{GL}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.
LA - eng
KW - Algebraic groups; Hilbert irreducibility
UR - http://eudml.org/doc/272251
ER -

References

top
  1. [1] J. Bernik, On groups and semigroups of matrices with spectra in a finitely generated field, Linear and Multilinear Algebra53 (2005), 259–267. Zbl1085.15020MR2160409
  2. [2] J. Bernik and J. Okniński, On semigroups of matrices with eigenvalue 1 in small dimension, Linear Algebra Appl.405 (2005), 67–73. Zbl1079.15017MR2148162
  3. [3] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Not.20 (1999), 1119–1140. Zbl0938.11031MR1728021
  4. [4] A. Borel, “Introduction aux groupes arithmétiques”, Hermann, Paris, 1969. Zbl0186.33202MR244260
  5. [5] A. Borel, “Linear Algebraic Groups”, 2nd Edition, GTM 126, Springer Verlag, 1997. Zbl0726.20030MR1102012
  6. [6] C. W. Curtis and I. Reiner, “Representation Theory of Finite Groups and Associative Algebras”, John Wiley & Sons, 1962. Zbl0634.20001
  7. [7] P. Dèbes, On the irreducibility of the polynomial P ( t m , Y ) , J. Number Theory, 42 (1992), 141–157. Zbl0770.12005MR1183373
  8. [8] R. Dvornicich and U. Zannier, Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J., to appear. Zbl1127.11040MR2350852
  9. [9] A. Ferretti and U. Zannier, Equations in the Hadamard ring of rational functions, Ann. Scuola Norm. Sup. Cl. Sci.6 (2007), 457–475. Zbl1150.11008MR2370269
  10. [10] D. Hilbert, Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 110 (1892), 104–129. Zbl24.0087.03JFM24.0087.03
  11. [11] S. Lang, “Fundamentals of Diophantine Geometry”, Springer Verlag, 1984. Zbl0528.14013MR715605
  12. [12] M. Laurent, Equations diophantiennes exponentielles et suites récurrentes linéaires II, J. Number Theory31 (1988), 24–53. Zbl0661.10027MR978098
  13. [13] D. W. Masser, Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc.311 (1989), 413–424. Zbl0673.14016MR974783
  14. [14] A. van der Poorten, Some facts that should be better known, especially about rational functions, In: “Number Theory and Applications (Banff, AB 1988)”, Kluwer Acad. Publ., Dordrecht (1989), 497-528. Zbl0687.10007MR1123092
  15. [15] G. Prasad and A. Rapinchuk, Existence of irreducible ℝ -regular elements in Zariski-dense subgroups, Math. Res. Lett.10 (2003), 21–32. Zbl1029.22020MR1960120
  16. [16] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Math. Res. Lett.12 (2005), 239–249. Zbl1072.22009MR2150880
  17. [17] R. Rumely, Notes on van der Poorten proof of the Hadamard quotient theorem II, In: “Séminaire de Théorie des Nombres de Paris 1986-87”, Progress in Mathematics, Birkhäuser, 1988. Zbl0661.10017MR990517
  18. [18] J.-P. Serre, “Lectures on the Mordell-Weil Theorem”, 3rd Edition, Vieweg-Verlag, 1997. Zbl0863.14013MR1757192
  19. [19] W. M. Schmidt, Linear Recurrence Sequences and Polynomial-Exponential Equations, In: “Diophantine Approximation”, F. Amoroso and U. Zannier (eds.), Proceedings of the C.I.M.E. Conference, Cetraro 2000, Springer LNM 1829, 2003. Zbl1034.11011MR2009831
  20. [20] U. Zannier, A proof of Pisot’s d -th root conjecture, Ann. of Math.151 (2000), 375–383. Zbl0995.11007MR1745008
  21. [21] U. Zannier, “Some Applications of Diophantine Approximations to Diophantine Equations”, Forum Editrice, Udine, 2003. Zbl0341.10017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.