On Hilbert’s irreducibility theorem for linear algebraic groups. (English) Zbl 1509.12001
Summary: We complete the proof of a version of Hilbert’s irreducibility theorem for linear algebraic groups over a finitely generated field of characteristic 0, originally announced in [T. Bröcker and T. tom Dieck, Representations of compact Lie groups. New York, etc.: Springer-Verlag. (1985; Zbl 0581.22009)]. We also give a sample application to the existence of generic elements in Zariski dense subgroup of linear groups [G. Prasad and A. S. Rapinchuk, Int. Math. Res. Not. 2001, No. 23, 1229–1242 (2001; Zbl 1057.22025)].
References:
| [1] | By (1), the modulo n 2 reduction of .⇡ ı / 1 .OEnç⇠/ does not contain R 2 =n 2 D F l -points. It follows that ⇡ 1 .OEnç⇠/ does not contain k 2 -points (the reason is the same as the last three sentences of Step 1). In view of Lemma 4.2, the proof is com-plete. |
| [2] | In the section, we prove the following result, which clearly implies Theorem 1.1. Theorem 5.1. Let G be a connected linear algebraic group over k, and let ⇢ G.k/ be a finitely generated Zariski dense subgroup. Let ⇡ W X ! G be a covering satisfying (PB). Then there is a finite-index coset of above which the fibres of ⇡ are irreducible. In particular, (HIT) holds for the pair .G; / over k. |
| [3] | L. BARY-SOROKER, A. FEHM and S. PETERSEN, On varieties of Hilbert type, Ann. Inst. Fourier (Grenoble) 64 (2014), 1893-1901. · Zbl 1359.12001 |
| [4] | E. BOMBIERI and W. GUBLER, “Heights in Diophantine Geometry”, New Math. Monogr., Vol. 4, Cambridge University Press, Cambridge, 2006. · Zbl 1115.11034 |
| [5] | T. BRÖCKER and T. T. DIECK, “Representations of Compact Lie Groups”, Springer Sci-ence & Business Media, Vol. 98, 2013. |
| [6] | J-L. COLLIOT-THÉLÈNE and J-J. SANSUC, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), 148-205. · Zbl 0597.14014 |
| [7] | P. CORVAJA, Rational fixed points for linear group actions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 561-597. · Zbl 1207.11067 |
| [8] | P. CORVAJA and U. ZANNIER, On the Hilbert property and the fundamental group of alge-braic varieties, Math. Z. 286 (2017), 579-602. · Zbl 1391.14044 |
| [9] | P. CORVAJA, J. L. DEMEIO, A. JAVANPEYKAR, D. LOMBARDO and U. ZANNIER, Hilbert’s irreducibility theorem for Abelian varieties, arXiv:2011.12840. |
| [10] | P. DÈBES, On the irreducibility of the polynomial P .t m ; Y /, J. Number Theory 42 (1992), 141-157. · Zbl 0770.12005 |
| [11] | R. DVORNICICH and U. ZANNIER, Cyclotomic Diophantine problems .Hilbert irreducibil-ity and invariant sets for polynomial maps/, Duke Math. J. 139 (2007), 527-554. · Zbl 1127.11040 |
| [12] | A. FERRETTI and U. ZANNIER, Equations in the Hadamard ring of rational functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 457-475. · Zbl 1150.11008 |
| [13] | B. FANTECHI, L. GOTTSCHE, L. ILLUSIE, S. L. KLEIMAN, N. NITSURE and A. VIS-TOLI, “Fundamental Algebraic Geometry: Grothendieck”s FGA Explained”, Mathematical Surveys and Monographs, Vol. 123, Amer. Math. Soc., 2005. · Zbl 1085.14001 |
| [14] | A. GORODNIK and A. NEVO, Splitting fields of elements in arithmetic groups, Math. Res. Lett. 18 (2011), 1281-1288. · Zbl 1372.22011 |
| [15] | A. GROTHENDIECK,Éléments de géométrie algébrique: IV.Étude locale des schémas et des morphismes de schémas, Troisième partie, Publ. Math. Inst. HautesÉtudes Sci. 28 (1966), 5-255. · Zbl 0144.19904 |
| [16] | G. P. HOCHSCHILD, “The Structure of Lie Groups”, Holden-Day, San Francisco, 1965. · Zbl 0131.02702 |
| [17] | F. JOUVE, E. KOWALSKI and D. ZYWINA, An explicit integral polynomial whose splitting field has Galois group W.E 8 /, J. Théor. Nombres Bordeaux 20 (2008), 761-782. · Zbl 1200.12003 |
| [18] | F. JOUVE, E. KOWALSKI and D. ZYWINA, Splitting fields of characteristic polynomials of random elements in arithmetic groups, Israel J. Math. 193 (2013), 263-307. · Zbl 1332.11098 |
| [19] | S. LANG, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555-563. · Zbl 0073.37901 |
| [20] | S. LANG and A. WEIL, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819-827. · Zbl 0058.27202 |
| [21] | S. LANG, “Fundamentals of Diophantine Geometry”, Springer Science & Business Media, 2013. |
| [22] | A. LUBOTZKY and L. ROSENZWEIG, The Galois group of random elements of linear groups, Amer. J. Math. 136 (2014), 1347-1383. · Zbl 1304.11125 |
| [23] | J. P. MURRE and S. ANANTHARAMAN, “Lectures on an Introduction to Grothendieck”s Theory of the Fundamental Group”, Lecture notes, Tata Institute of Fundamental Research, Bombay, 1967. · Zbl 0198.26202 |
| [24] | M. V. NORI, On subgroups of GL n .F p /, Invent. Math. 88 (1987), 257-275. · Zbl 0632.20030 |
| [25] | G. PRASAD and A. S. RAPINCHUK, Irreducible tori in semisimple groups, Int. Math. Res. Not. IMRN 2001 (2001), 1229-1242. · Zbl 1057.22025 |
| [26] | G. PRASAD and A. S. RAPINCHUK, Existence of irreducible R-regular elements in Zariski-dense subgroups, Math. Res. Lett. 10 (2003), 21-32. · Zbl 1029.22020 |
| [27] | G. PRASAD and A. S. RAPINCHUK, Generic elements of a Zariski-dense subgroup form an open subset, Trans. Moscow Math. Soc. 78 (2017), 299-314. · Zbl 1423.22015 |
| [28] | J-P. SERRE, “Topics in Galois Theory”, with notes by Henri Darmon 4.3, Research Notes in Mathematics, Vol. 1, AK Peters, Ltd., Wellesley, MA, 2008. |
| [29] | J-P. SERRE, “Lectures on N X .p/”, Research Notes in Mathematics, Vol. 11, Chapman & Hall/CRC, 2012. · Zbl 1238.11001 |
| [30] | U. ZANNIER, A proof of Pisot’s d -th root conjecture, Ann. of Math. (2) 151 (2000), 375-383. · Zbl 0995.11007 |
| [31] | U. ZANNIER, Hilbert irreducibility above algebraic groups, Duke Math. J. 153 (2010), 397-425. School of Mathematical Sciences Peking University Beijing, P. R. of China China liufei54@pku.edu.cn · Zbl 1208.11080 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
