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  • Corentin Perret-Gentil (a1)


We show that integral monodromy groups of Kloosterman $\ell$ -adic sheaves of rank $n\geqslant 2$ on $\mathbb{G}_{m}/\mathbb{F}_{q}$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz’s results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_{q})$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.



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Current address: Centre de recherches mathématiques, Université de Montréal, Case postale 6128, Montréal QC H3C 3J7, Canada



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1. Aschbacher, M., On the maximal subgroups of the finite classical groups. Invent. Math. 76(3) 1984, 469514.
2. Carter, R. W., Simple Groups of Lie Type, John Wiley & Sons (London, New York, Sydney, Toronto, 1972).
3. Deligne, P., La conjecture de Weil. I. Publ. Math. Inst. Hautes Études Sci. 43(1) 1974, 273307.
4. Deligne, P., Cohomologie Étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA $\mathit{4}\frac{\mathit{1}}{\mathit{2}}$ (Lecture Notes in Mathematics 569), Springer (Berlin, Heidelberg, 1977).
5. Deligne, P., La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. 52(1) 1980, 137252.
6. Fisher, B., Kloosterman sums as algebraic integers. Math. Ann. 301(1) 1995, 485505.
7. Fu, L., Calculation of -adic local Fourier transformations. Manuscripta Math. 133(3–4) 2010, 409464.
8. Fulton, W. and Harris, J., Representation Theory (Graduate Texts in Mathematics 129 ), Springer (New York, 1991).
9. Gorenstein, D., Finite Simple Groups: An Introduction to their Classification (University Series in Mathematics), Springer (New York, 1982).
10. Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups (Mathematical Surveys and Monographs 40 ), American Mathematical Society (1994).
11. Gow, R. and Tamburini, M. C., Generation of SL(n, p) by two Jordan block matrices. Boll. dell’Unione Mat. Ital. 7(6A) 1992, 346357.
12. Hall, C., Big symplectic or orthogonal monodromy modulo  . Duke Math. J. 141(1) 2008, 179203.
13. Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics 9 ), Springer (New York, 1980).
14. Iwaniec, H. and Kowalski, E., Analytic Number Theory (Colloquium Publications), American Mathematical Society (Providence, RI, 2004).
15. Kantor, W. M. and Lubotzky, A., The probability of generating a finite classical group. Geom. Dedicata 36(1) 1990, 6787.
16. Katz, N. M., Gauss Sums, Kloosterman Sums, and Monodromy Groups (Annals of Mathematics Studies 116 ), Princeton University Press (Princeton, NJ, 1988).
17. Katz, N. M., Exponential Sums and Differential Equations (Annals of Mathematics Studies 124 ), Princeton University Press (Princeton, NJ, 1990).
18. Katz, N. M., Report on the irreducibility of L-functions. In Number Theory, Analysis and Geometry, (eds Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K. and Tate, J.), Springer (New York, 2012), 321353.
19. Katz, N. M. and Sarnak, P., Random Matrices, Frobenius Eigenvalues and Monodromy (Colloquium Publications 45 ), American Mathematical Society (Providence, RI, 1991).
20. Kleidman, P. B. and Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups (London Mathematical Society Lecture Notes 129 ), Cambridge University Press (Cambridge, 1990).
21. Kloosterman, H., On the representation of numbers in the form ax 2 + by 2 + cz 2 + dt 2 . Acta Math. 49(3) 1927, 407464.
22. Kowalski, E., On the rank of quadratic twists of elliptic curves over function fields. Int. J. Number Theory 2(2) 2006, 267288.
23. Kowalski, E., The large sieve, monodromy and zeta functions of curves. J. Reine Angew. Math. 601 2006, 2969.
24. Kowalski, E., Weil numbers generated by other Weil numbers and torsion field of abelian varieties. J. Lond. Math. Soc. (2) 74(2) 2006, 273288.
25. Kowalski, E., The Large Sieve and its Applications: Arithmetic Geometry, Random Walks and Discrete Groups (Cambridge Tracts in Mathematics 175 ), Cambridge University Press (Cambridge, 2008).
26. Kowalski, E., Michel, P. and Sawin, W., Bilinear forms with Kloosterman sums and applications. Ann. of Math. (2) 186(2) 2017, 413500.
27. Landazuri, V. and Seitz, G. M., On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32(2) 1974, 418443.
28. Larsen, M., Maximality of Galois actions for compatible systems. Duke Math. J. 80(3) 1995, 601630.
29. Larsen, M. and Pink, R., Finite subgroups of algebraic groups. J. Amer. Math. Soc. 24(4) 2011, 11051158.
30. Liebeck, M. W., On the orders of maximal subgroups of the finite classical groups. Proc. Lond. Math. Soc. (3) 3(3) 1985, 426446.
31. Liebeck, M. W. and Seitz, G. M., On the subgroup structure of classical groups. Invent. Math. 134(2) 1998, 427453.
32. Malle, G. and Testerman, D., Linear Algebraic Groups and Finite Groups of Lie Type (Cambridge Studies in Advanced Mathematics 133 ), Cambridge University Press (Cambridge, 2011).
33. Nori, M. V., On subgroups of GL n (F p ). Invent. Math. 88(2) 1987, 257275.
34. Perret-Gentil, C., Probabilistic aspects of short sums of trace functions over finite fields. PhD Thesis, ETH Zürich, 2016.
35. Robinson, D., A Course in the Theory of Groups (Graduate Texts in Mathematics 80 ), Springer (1996).
36. Saxl, J. and Seitz, G. M., Subgroups of algebraic groups containing regular unipotent elements. J. Lond. Math. Soc. (2) 55(02) 1997, 370386.
37. Seitz, G. M. and Testerman, D. M., Extending morphisms from finite to algebraic groups. J. Algebra 131(2) 1990, 559574.
38. Serre, J.-P., Abelian -Adic Representations and Elliptic Curves (Research Notes in Mathematics 7 ), Addison-Wesley (Reading, MA, 1989).
39. Suprunenko, I. D., Irreducible representations of simple algebraic groups containing matrices with big Jordan blocks. Proc. Lond. Math. Soc. (3) 3(2) 1995, 281332.
40. Testerman, D. and Zalesski, A., Irreducibility in algebraic groups and regular unipotent elements. Proc. Amer. Math. Soc. 141(1) 2013, 1328.
41. Wagner, A., The faithful linear representations of least degree of S n and A n over a field of odd characteristics. Math. Z. 154 1977, 103114.
42. Wan, D., Minimal polynomials and distinctness of Kloosterman sums. Finite Fields Appl. 1(2) 1995, 189203.
43. Washington, L. C., Introduction to Cyclotomic Fields (Graduate Texts in Mathematics 83 ), Springer (New York, 1997).
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  • Corentin Perret-Gentil (a1)


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