Abstract
We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as the large sieve, which we had already used, we introduce some probabilistic ideas (large deviation estimates for finite Markov chains) and the general case involves a more precise understanding of the way Frobenius conjugacy classes are computed for such splitting fields (which is related to a map between regular elements of a finite group of Lie type and conjugacy classes in the Weyl group which had been considered earlier by Carter and Fulman for other purposes; we show in particular that the values of this map are equidistributed).
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R. N. Bhattacharya and E C. Waymire, Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1990.
A. Borel, Linear Algebraic Groups, 2nd edition, Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2(F p), Annals of Mathematics 167 (2008), 625–642.
E. Breuillard, B. Green and T. Tao, Approximate subgroups of linear groups, Geometric and Functional Analysis 21 (2011), 774–819.
R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985.
R. W. Carter, Semisimple conjugacy classes and classes in the Weyl group, Journal of Algebra 260 (2003), 99–110.
L. Clozel, Démonstration de la conjecture τ, Inventiones Mathematicae 151 (2003), 297–328.
M. Demazure, Schémas en groupes réductifs, Bulletin de la Société Mathématique de France 93 (1965), 369–413.
J. Fulman, Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems, Journal of Algebra 243 (2001), 96–122.
A. Gorodnik and A. Nevo, Splitting fields of elements in arithmetic groups, Mathematical Research Letters 18 (2011), 1281–1288.
A. Gorodnik and A. Nevo, The Ergodic Theory of Lattice Subgroups, Annals of Mathematics Studies, Vol. 172, Princeton University Press, Princeton, NJ, 2010.
H. Helfgott, Growth and generation in SL 2(Z /p Z), Annals of Mathematics 167 (2008), 601–623.
E. Hrushovski and A. Pillay, Definable subgroups of algebraic groups over finite fields, Journal für die Reine und Angewandte Mathematik 462 (1995), 69–91.
H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, Vol. 53, American Mathematical Society, Providence, RI, 2004.
F. Jouve, E. Kowalski and D. Zywina, An explicit integral polynomial whose splitting field has Galois group W(E 8), Journal de Théorie des Nombres de Bordeaux 20 (2008), 761–782.
F. Jouve, The large sieve and random walks on left cosets of arithmetic groups, Commentarii Mathematici Helvetici 85 (2010), 647–704.
N. M. Katz, Factoring polynomials in finite fields: an application of Lang-Weil to a problem in graph theory, Mathematische Annalen 286 (1990), 625–637.
E. Kowalski, The Large Sieve and its Applications: Arithmetic Geometry, Random Walks, Discrete Groups, Cambridge Tracts in Mathematics, Vol. 175, Cambridge University Press, Cambridge, 2008.
P. Lezaud, Chernoff-type bound for finite Markov chains, The Annals of Applied Probability 8 (1998), 849–867.
C. R. Matthews, L. N. Vaserstein and B. Weisfeiler, Congruence properties of Zariskidense subgroups, Proceedings of the London Mathematical Society 14 (1982), 514–532.
M. V. Nori, On subgroups of GL n(F p), Inventiones Mathematicae 88 (1987), 257–275.
V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, New York, 1994.
G. Prasad and A. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publications Mathématiques. Institut de Hautes Études Scientifiques 109 (2009), 113–184.
L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, preprint (2010), arXiv:1005.1858v1
A. Salehi Golsefidy and P. Varjú, Expansion in perfect groups, preprint (2010), arXiv:1108.4900v2.
L. Saloff-Coste, Random walks on finite groups, in Probability on discrete structures I, Encyclopaedia of Mathematical Sciences, Vol. 110, Springer, Berlin, 2004, pp. 263–346.
Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin, 1962/1964.
T. A. Springer, Linear Algebraic Groups, 2nd edition, Progress in Mathematics, Vol. 9, Birkhaüser, Basel, 1998.
R. Steinberg, Regular elements of semisimple algebraic groups, Publications Mathématiques. Institut de Hautes Études Scientifiques 25 (1965), 49–80.
B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups, Annals of Mathematics 120 (1984), 271–315.
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Jouve, F., Kowalski, E. & Zywina, D. Splitting fields of characteristic polynomials of random elements in arithmetic groups. Isr. J. Math. 193, 263–307 (2013). https://doi.org/10.1007/s11856-012-0117-x
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DOI: https://doi.org/10.1007/s11856-012-0117-x