Rational fixed points for linear group actions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, p. 561-597
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:VG, 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 ΓG(κ); namely, there must exist an unramified covering p:G ˜G and a map θ:G ˜V such that πθ=p. In the case κ=, 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.
Classification:  11G35,  14G25
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {4},
     year = {2007},
     pages = {561-597},
     zbl = {1207.11067},
     mrnumber = {2394411},
     language = {en},
     url = {http://www.numdam.org/item/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, Série 5, Tome 6 (2007) no. 4, pp. 561-597. http://www.numdam.org/item/ASNSP_2007_5_6_4_561_0/

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