We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group , an affine variety and a finite map , all defined over a finitely generated field of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set contains a Zariski dense sub-semigroup ; namely, there must exist an unramified covering and a map such that . In the case , 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 admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.
@article{ASNSP_2007_5_6_4_561_0, author = {Corvaja, Pietro}, title = {Rational fixed points for linear group actions}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {561--597}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {4}, year = {2007}, mrnumber = {2394411}, zbl = {1207.11067}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_4_561_0/} }
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 SP - 561 EP - 597 VL - 6 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_4_561_0/ LA - en ID - ASNSP_2007_5_6_4_561_0 ER -
%0 Journal Article %A Corvaja, Pietro %T Rational fixed points for linear group actions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 561-597 %V 6 %N 4 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_4_561_0/ %G en %F ASNSP_2007_5_6_4_561_0
Corvaja, Pietro. Rational fixed points for linear group actions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 4, pp. 561-597. http://archive.numdam.org/item/ASNSP_2007_5_6_4_561_0/
[1] On groups and semigroups of matrices with spectra in a finitely generated field, Linear and Multilinear Algebra 53 (2005), 259-267. | MR | Zbl
,[2] On semigroups of matrices with eigenvalue in small dimension, Linear Algebra Appl. 405 (2005), 67-73. | MR | Zbl
and ,[3] Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Not. 20 (1999), 1119-1140. | MR | Zbl
, and ,[4] “Introduction aux groupes arithmétiques”, Hermann, Paris, 1969. | MR | Zbl
,[5] “Linear Algebraic Groups”, 2nd Edition, GTM 126, Springer Verlag, 1997. | MR | Zbl
,[6] “Representation Theory of Finite Groups and Associative Algebras”, John Wiley & Sons, 1962. | Zbl
and ,[7] On the irreducibility of the polynomial , J. Number Theory, 42 (1992), 141-157. | MR | Zbl
,[8] Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J., to appear. | MR | Zbl
and ,[9] Equations in the Hadamard ring of rational functions, Ann. Scuola Norm. Sup. Cl. Sci. 6 (2007), 457-475. | EuDML | Numdam | MR | Zbl
and ,[10] Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 110 (1892), 104-129. | JFM
,[11] “Fundamentals of Diophantine Geometry”, Springer Verlag, 1984. | MR | Zbl
,[12] Equations diophantiennes exponentielles et suites récurrentes linéaires II, J. Number Theory 31 (1988), 24-53. | MR | Zbl
,[13] Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc. 311 (1989), 413-424. | MR | Zbl
,[14] 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. | MR | Zbl
,[15] Existence of irreducible -regular elements in Zariski-dense subgroups, Math. Res. Lett. 10 (2003), 21-32. | MR | Zbl
and ,[16] Zariski-dense subgroups and transcendental number theory, Math. Res. Lett. 12 (2005), 239-249. | MR | Zbl
and ,[17] 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. | MR | Zbl
,[18] “Lectures on the Mordell-Weil Theorem”, 3rd Edition, Vieweg-Verlag, 1997. | MR | Zbl
,[19] 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. | MR | Zbl
,[20] A proof of Pisot’s -th root conjecture, Ann. of Math. 151 (2000), 375-383. | MR | Zbl
,[21] “Some Applications of Diophantine Approximations to Diophantine Equations”, Forum Editrice, Udine, 2003. | Zbl
,