Distribution questions for trace functions with values in cyclotomic integers and their reductions
HTML articles powered by AMS MathViewer
- by Corentin Perret-Gentil PDF
- Trans. Amer. Math. Soc. 371 (2019), 4585-4629
Abstract:
We consider $\ell$-adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.References
- Jean Bourgain and Mei-Chu Chang, A Gauss sum estimate in arbitrary finite fields, C. R. Math. Acad. Sci. Paris 342 (2006), no. 9, 643–646 (English, with English and French summaries). MR 2225868, DOI 10.1016/j.crma.2006.01.022
- Jean Bourgain and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Math. Acad. Sci. Paris 337 (2003), no. 2, 75–80 (English, with English and French summaries). MR 1998834, DOI 10.1016/S1631-073X(03)00281-4
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
- Benji Fisher, Kloosterman sums as algebraic integers, Math. Ann. 301 (1995), no. 3, 485–505. MR 1324522, DOI 10.1007/BF01446641
- Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel, Trace functions over finite fields and their applications, Colloquium De Giorgi 2013 and 2014, Colloquia, vol. 5, Ed. Norm., Pisa, 2014, pp. 7–35. MR 3379177
- Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel, Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal. 25 (2015), no. 2, 580–657. MR 3334236, DOI 10.1007/s00039-015-0310-2
- Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel, A study in sums of products, Philos. Trans. Roy. Soc. A 373 (2015), no. 2040, 20140309, 26. MR 3338119, DOI 10.1098/rsta.2014.0309
- M. Z. Garaev, An explicit sum-product estimate in $\Bbb F_p$, Int. Math. Res. Not. IMRN 11 (2007), Art. ID rnm035, 11. MR 2344270
- Daniel Gorenstein, Finite simple groups, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR 698782, DOI 10.1007/978-1-4684-8497-7
- Chris Hall, Big symplectic or orthogonal monodromy modulo $l$, Duke Math. J. 141 (2008), no. 1, 179–203. MR 2372151, DOI 10.1215/S0012-7094-08-14115-8
- D. R. Heath-Brown and S. Konyagin, New bounds for Gauss sums derived from $k\textrm {th}$ powers, and for Heilbronn’s exponential sum, Q. J. Math. 51 (2000), no. 2, 221–235. MR 1765792, DOI 10.1093/qjmath/51.2.221
- Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
- Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536, DOI 10.1515/9781400882434
- Dae San Kim, Gauss sums for general and special linear groups over a finite field, Arch. Math. (Basel) 69 (1997), no. 4, 297–304. MR 1466823, DOI 10.1007/s000130050124
- Dae San Kim, Gauss sums for $\textrm {O}^-(2n,q)$, Acta Arith. 80 (1997), no. 4, 343–365. MR 1450928, DOI 10.4064/aa-80-4-343-365
- Dae San Kim, Gauss sums for $\textrm {O}(2n+1,q)$, Finite Fields Appl. 4 (1998), no. 1, 62–86. MR 1612080, DOI 10.1006/ffta.1997.0202
- Dae San Kim, Gauss sums for symplectic groups over a finite field, Monatsh. Math. 126 (1998), no. 1, 55–71. MR 1633263, DOI 10.1007/BF01312455
- Dae San Kim and In-Sok Lee, Gauss sums for $\textrm {O}^{+}(2n,q)$, Acta Arith. 78 (1996), no. 1, 75–89. MR 1425001, DOI 10.4064/aa-78-1-75-89
- N. M. Korobov, Exponential sums and their applications, Mathematics and its Applications (Soviet Series), vol. 80, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the 1989 Russian original by Yu. N. Shakhov. MR 1162539, DOI 10.1007/978-94-015-8032-8
- E. Kowalski, On the rank of quadratic twists of elliptic curves over function fields, Int. J. Number Theory 2 (2006), no. 2, 267–288. MR 2240230, DOI 10.1142/S1793042106000528
- E. Kowalski, The large sieve and its applications, Cambridge Tracts in Mathematics, vol. 175, Cambridge University Press, Cambridge, 2008. Arithmetic geometry, random walks and discrete groups. MR 2426239, DOI 10.1017/CBO9780511542947
- E. Kowalski, Explicit multiplicative combinatorics, unpublished note, https://people.math.ethz.ch/~kowalski/combinatorics.pdf, October 2011.
- Lars Kindler and Kay Rülling, Introductory course on $\ell$-adic sheaves and their ramification theory on curves, September 2015, http://arxiv.org/abs/1409.6899.
- Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828, DOI 10.1090/coll/045
- Youness Lamzouri, The distribution of short character sums, Math. Proc. Cambridge Philos. Soc. 155 (2013), no. 2, 207–218. MR 3091515, DOI 10.1017/S0305004113000170
- Youness Lamzouri and Alexandru Zaharescu, Randomness of character sums modulo $m$, J. Number Theory 132 (2012), no. 12, 2779–2792. MR 2965191, DOI 10.1016/j.jnt.2012.05.024
- Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737, DOI 10.1017/CBO9780511994777
- Kit-Ho Mak and Alexandru Zaharescu, On the distribution of the number of points on a family of curves over finite fields, J. Number Theory 140 (2014), 277–298. MR 3181658, DOI 10.1016/j.jnt.2014.01.012
- Corentin Perret-Gentil, Integral monodromy groups of Kloosterman sheaves, Mathematika 64 (2018), no. 3, 652–678. MR 3826481, DOI 10.1112/s0025579318000189
- Corentin Perret-Gentil, Probabilistic aspects of short sums of trace functions over finite fields, Ph.D. thesis, ETH Zürich, 2016.
- Corentin Perret-Gentil, Gaussian distribution of short sums of trace functions over finite fields, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 3, 385–422. MR 3708517, DOI 10.1017/S0305004117000020
- D. H. J. Polymath, New equidistribution estimates of Zhang type, Algebra Number Theory 8 (2014), no. 9, 2067–2199. MR 3294387, DOI 10.2140/ant.2014.8.2067
- Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 457455, DOI 10.2307/2373815
- Da Qing Wan, Minimal polynomials and distinctness of Kloosterman sums, Finite Fields Appl. 1 (1995), no. 2, 189–203. Special issue dedicated to Leonard Carlitz. MR 1337743, DOI 10.1006/ffta.1995.1015
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Additional Information
- Corentin Perret-Gentil
- Affiliation: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
- Address at time of publication: Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec H3C 3J7, Canada
- Email: corentin.perretgentil@gmail.com
- Received by editor(s): October 27, 2016
- Received by editor(s) in revised form: June 20, 2017
- Published electronically: August 9, 2018
- Additional Notes: This work was partially supported by DFG-SNF lead agency program grant 200021L_153647. The final corrections were made while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester, supported by the National Science Foundation under grant No. DMS-1440140. The results also appear in the author’s PhD thesis “Probabilistic aspects of short sums of trace functions over finite fields”.
- © Copyright 2018 by Corentin Perret-Gentil
- Journal: Trans. Amer. Math. Soc. 371 (2019), 4585-4629
- MSC (2010): Primary 11L05, 11T24, 11N64, 14F20, 60G50
- DOI: https://doi.org/10.1090/tran/7333
- MathSciNet review: 3934462