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RIGID LOCAL SYSTEMS ON $\mathbb{A}^{1}$ WITH FINITE MONODROMY

Part of: Differential equations in the complex domain Finite fields and commutative rings (number-theoretic aspects) Permutation groups

Published online by Cambridge University Press:  01 August 2018

Nicholas M. Katz
Nicholas M. Katz*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, U.S.A. email [email protected]
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Abstract

We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on $\mathbb{A}^{1}$ whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.

MSC classification

Primary: 11T23: Exponential sums
Secondary: 20B05: General theory for finite groups 34M35: Singularities, monodromy, local behavior of solutions, normal forms
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 785 - 846
Copyright
Copyright © University College London 2018 

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References

Berndt, B. C., Evans, R. J. and Williams, K. S., Gauss and Jacobi Sums (Canadian Mathematical Society Series of Monographs and Advanced Texts), Wiley (New York, 1998).Google Scholar
Brauer, R., Über endliche lineare Gruppen von Primzahlgrad. Math. Ann. 169 1967, 73–96.Google Scholar
Brauer, R., On the order of finite projective groups in a given dimension. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969, 103–106.Google Scholar
Deligne, P., La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. 52 1980, 137–252.Google Scholar
Ennola, V., On the characters of the finite unitary groups. Ann. Acad. Sci. Fenn. Math. 323 1963.Google Scholar
Evans, R., Notes available at http://math.ucsd.edu/∼revans/katzconj.pdf, 2017.Google Scholar
Feit, W., Groups which have a faithful representation of dimension < p - 1. Trans. Amer. Math. Soc. 112 1964, 287–303.Google Scholar
Feit, W. and Thompson, J. G., Groups which have a faithful representation of degree less than (p - 1)/2. Pacific J. Math. 11 1961, 1257–1262.Google Scholar
Garrett, P., Heisenberg groups over finite fields. 2014–2015 lecture notes available atwww.math.umn.edu/∼garrett/m/repns/notes_2014-15/05_finite_heisenberg_ssw.pdf.Google Scholar
Gross, B. H., Rigid local systems on G m with finite monodromy. Adv. Math. 224(6) 2010, 2531–2543.Google Scholar
Grothendieck, A., Formule de Lefschetz et rationalité des fonctions L. (Séminaire Bourbaki 9 , Exp. No. 279), Soc. Math. France (Paris, 1995), 41–55.Google Scholar
Grove, L., Classical Groups and Geometric Algebra (Graduate Studies in Mathematics 39 ), American Mathematical Society (Providence, RI, 2001).Google Scholar
Guralnick, R., Maagard, K. and Tiep, P. H., Symmetric and alternating powers of Weil representations of finite symplectic groups. Bull. Inst. Math. Acad. Sin. (N.S.), to appear.Google Scholar
Hiss, G. and Malle, G., Low-dimensional representations of special unitary groups. J. Algebra 236(2) 2001, 745–767.Google Scholar
Katz, N., On the monodromy groups attached to certain families of exponential sums. Duke Math. J. 34(1) 1987, 41–56.Google Scholar
Katz, N., Gauss Sums, Kloosterman Sums, and Monodromy Groups (Annals of Mathematics Studies 116 ), Princeton University Press (Princeton, NJ, 1988).Google Scholar
Katz, N., Exponential Sums and Differential Equations (Annals of Mathematics Studies 124 ), Princeton University Press (Princeton, NJ, 1990).Google Scholar
Katz, N., Rigid Local Systems (Annals of Mathematics Studies 139 ), Princeton University Press (Princeton, NJ, 1996).Google Scholar
Katz, N., L-functions and monodromy: four lectures on Weil II. Adv. Math. 160(1) 2001, 81–132.Google Scholar
Katz, N., Notes on G2, determinants, and equidistribution. Finite Fields Appl. 10(2) 2004, 221–269.Google Scholar
Katz, N., Moments, Monodromy, and Perversity: A Diophantine Perspective (Annals of Mathematics Studies 159 ), Princeton University Press (Princeton, NJ, 2005).Google Scholar
Katz, N., G 2 and hypergeometric sheaves. Finite Fields Appl. 13(2) 2007, 175–223.Google Scholar
Kubert, D., Lectures at Princeton University, 1986.Google Scholar
Laumon, G., Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. Publ. Math. Inst. Hautes Études Sci. 65 1987, 131–210.Google Scholar
Lusztig, G., Coxeter orbits and eigenspaces of Frobenius. Invent. Math. 38 1976, 101–159.Google Scholar
Pink, R., Lectures at Princeton University, 1986.Google Scholar
Raynaud, M., Revêtements de la droite affine en caractéristique p > 0 et conjecture d’Abhyankar. Invent. Math. 116(1–3) 1994, 425–462.+0+et+conjecture+d’Abhyankar.+Invent.+Math.+116(1–3)+1994,+425–462.>Google Scholar
Tuan, H.-F., On groups whose orders contain a prime number to the first power. Ann. of Math. (2) 45 1944, 110–140.Google Scholar