Tunisian Journal of Mathematics
- Tunisian J. Math.
- Volume 1, Number 3 (2019), 295-320.
Rigid local systems and alternating groups
Robert M. Guralnick, Nicholas M. Katz, and Pham Huu Tiep
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Abstract
We show that some very simple to write one parameter families of exponential sums on the affine line in characteristic have alternating groups as their geometric monodromy groups.
Article information
Source
Tunisian J. Math., Volume 1, Number 3 (2019), 295-320.
Dates
Received: 5 October 2017
Revised: 3 April 2018
Accepted: 22 April 2018
First available in Project Euclid: 15 December 2018
Permanent link to this document
https://projecteuclid.org/euclid.tunis/1544842815
Digital Object Identifier
doi:10.2140/tunis.2019.1.295
Mathematical Reviews number (MathSciNet)
MR3907742
Zentralblatt MATH identifier
07027457
Subjects
Primary: 11T23: Exponential sums 20D05: Finite simple groups and their classification
Keywords
rigid local system monodromy alternating group
Citation
Guralnick, Robert M.; Katz, Nicholas M.; Tiep, Pham Huu. Rigid local systems and alternating groups. Tunisian J. Math. 1 (2019), no. 3, 295--320. doi:10.2140/tunis.2019.1.295. https://projecteuclid.org/euclid.tunis/1544842815
References
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups: maximal subgroups and ordinary characters for simple groups, Oxford University Press, Eynsham, 1985.
- C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics XI, Interscience, New York, 1962.
- P. Deligne, “La conjecture de Weil, II”, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252. Mathematical Reviews (MathSciNet): MR601520
Zentralblatt MATH: 0456.14014
Digital Object Identifier: doi:10.1007/BF02684780 - F. Digne and J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts 21, Cambridge University Press, 1991.
- L. Fu, “Calculation of $\ell$-adic local Fourier transformations”, Manuscripta Math. 133:3-4 (2010), 409–464. Mathematical Reviews (MathSciNet): MR2729262
Zentralblatt MATH: 1206.14035
Digital Object Identifier: doi:10.1007/s00229-010-0377-x - W. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics 129, Springer, 1991.
- GAP –- Groups, Algorithms, and Programming, Version 4.4, The GAP Group, 2004, https://www.gap-system.org. URL: Link to item
- D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, vol. 3, Mathematical Surveys and Monographs 40, American Mathematical Society, Providence, RI, 1998.
- B. H. Gross, “Rigid local systems on $\mathbb G_m$ with finite monodromy”, Adv. Math. 224:6 (2010), 2531–2543. Mathematical Reviews (MathSciNet): MR2652215
Zentralblatt MATH: 1193.22001
Digital Object Identifier: doi:10.1016/j.aim.2010.02.008 - A. Grothendieck, “Formule de Lefschetz et rationalité des fonctions $L$”, pp. 31–45 in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968.
- R. M. Guralnick and P. H. Tiep, “Symmetric powers and a problem of Kollár and Larsen”, Invent. Math. 174:3 (2008), 505–554. Mathematical Reviews (MathSciNet): MR2453600
Zentralblatt MATH: 1245.20058
Digital Object Identifier: doi:10.1007/s00222-008-0140-z - R. Howe, “Another look at the local $\theta$-correspondence for an unramified dual pair”, pp. 93–124 in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, I (Ramat Aviv, Israel, 1989), Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990.
- C. Jordan, “Recherches sur les substitutions”, Journal de Mathématiques Pures et Appliquées 17 (1872), 351–367. Zentralblatt MATH: 04.0055.01
- N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies 116, Princeton University Press, 1988.
- N. M. Katz, “Perversity and exponential sums”, pp. 209–259 in Algebraic number theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, 1989.
- N. M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies 124, Princeton University Press, 1990.
- N. M. Katz, Rigid local systems, Annals of Mathematics Studies 139, Princeton University Press, 1996.
- N. M. Katz, “Notes on $G_2$, determinants, and equidistribution”, Finite Fields Appl. 10:2 (2004), 221–269. Mathematical Reviews (MathSciNet): MR2045016
Zentralblatt MATH: 1065.11060
Digital Object Identifier: doi:10.1016/j.ffa.2003.11.002 - N. M. Katz, Moments, monodromy, and perversity: a Diophantine perspective, Annals of Mathematics Studies 159, Princeton University Press, 2005.
- N. Katz, “Rigid local systems on $\smash{\A^1}$ with finite monodromy”, preprint, 2018, http://www.math.princeton.edu/~nmk/gpconj114.pdf. With an appendix by P. H. Tiep. To appear in Mathematika. URL: Link to item
Mathematical Reviews (MathSciNet): MR3867322
Digital Object Identifier: doi:10.1112/S0025579318000268 - N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications 45, American Mathematical Society, Providence, RI, 1999.
- F. Lübeck, “Character degrees and their multiplicities for some groups of Lie type of rank $< 9$”, web site, 2007, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html. URL: Link to item
- D. I. Panyushev, “Weight multiplicity free representations, $\mathfrak g$-endomorphism algebras, and Dynkin polynomials”, J. London Math. Soc. $(2)$ 69:2 (2004), 273–290. Mathematical Reviews (MathSciNet): MR2040603
Zentralblatt MATH: 1052.17001
Digital Object Identifier: doi:10.1112/S0024610703004873 - R. Rasala, “On the minimal degrees of characters of $S\sb{n}$”, J. Algebra 45:1 (1977), 132–181. Mathematical Reviews (MathSciNet): MR0427445
Zentralblatt MATH: 0348.20009
Digital Object Identifier: doi:10.1016/0021-8693(77)90366-0 - M. Raynaud, “Revêtements de la droite affine en caractéristique $p>0$ et conjecture d'Abhyankar”, Invent. Math. 116:1-3 (1994), 425–462. Mathematical Reviews (MathSciNet): MR1253200
Zentralblatt MATH: 0798.14013
Digital Object Identifier: doi:10.1007/BF01231568 - J.-P. Serre, “On a theorem of Jordan”, Bull. Amer. Math. Soc. $($N.S.$)$ 40:4 (2003), 429–440. Mathematical Reviews (MathSciNet): MR1997347
Zentralblatt MATH: 1047.11045
Digital Object Identifier: doi:10.1090/S0273-0979-03-00992-3 - P. H. Tiep, “Low dimensional representations of finite quasisimple groups”, pp. 277–294 in Groups, combinatorics & geometry (Durham, NC, 2001), edited by A. A. Ivanov et al., World Sci. Publ., River Edge, NJ, 2003.
- P. H. Tiep and A. E. Zalesskii, “Minimal characters of the finite classical groups”, Comm. Algebra 24:6 (1996), 2093–2167. Mathematical Reviews (MathSciNet): MR1386030
Zentralblatt MATH: 0901.20031
Digital Object Identifier: doi:10.1080/00927879608825690
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