Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-29T19:01:24.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian distribution of short sums of trace functions over finite fields

Published online by Cambridge University Press:  20 March 2017

CORENTIN PERRET–GENTIL
CORENTIN PERRET–GENTIL*
Affiliation:
ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 385 - 422
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BRR86] Bhattacharya, R. N. and Ranga Rao, R. Normal Approximation and Asymptotic Expansions (Robert E. Krieger Publishing Co., 1986). Reprint of the 1976 original.Google Scholar
[DE52] Davenport, H. and Erdős, P. The distribution of quadratic and higher residues. Publ. Math. Debrecen 2 (1952), 252–265.Google Scholar
[Del77] Deligne, P. Cohomologie étale, séminaire de géométrie algébrique du Bois-Marie SGA 4 $\frac{1}{2}$ , Lecture Notes in Math. vol 569 (Springer, 1977).Google Scholar
[Del80] Deligne, P. La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. 52 (1) (1980), 137–252.Google Scholar
[DS94] Diaconis, P. and Shahshahani, M. On the eigenvalues of random matrices. J. Appl. Probab. 31 (1994), 49–62.Google Scholar
[FH91] Fulton, W. and Harris, J. Representation theory. Graduate Texts in Mathematics, vol. 129 (Springer, 1991).Google Scholar
[FKM14a] Fouvry, É., Kowalski, E. and Michel, P. Trace functions over finite fields and applications. https://people.math.ethz.ch/~kowalski/elements.pdf (December 2014).Google Scholar
[FKM14b] Fouvry, É., Kowalski, E. and Michel, P. Trace functions over finite fields and their applications. In Colloquium De Giorgi 2013 and 2014. Colloquia, vol. 5 (Ed. Norm., Pisa, 2014), pp. 7–35.Google Scholar
[FKM15a] Fouvry, É., Kowalski, E. and Michel, P. Algebraic twists of modular forms and Hecke orbits. Geom. Funct. Anal. 25 (2) (2015), 580–657.Google Scholar
[FKM15b] Fouvry, É., Kowalski, E. and Michel, P. A study in sums of products. Philos. Trans. A 373 (2040) (2015).Google Scholar
[FM02] Fouvry, É. and Michel, P. A la recherche de petites sommes d'exponentielles. Ann. Inst. Fourier (Grenoble) 52 (1) (2002), 47–80.Google Scholar
[FM03] Fouvry, É. and Michel, P. Sommes de modules de sommes d'exponentielles. Pacific J. Math. 209 (2) (2003).Google Scholar
[Gut05] Gut, A. Probability: a Graduate Course. Springer texts in statistics (Springer, 2005).Google Scholar
[Hal08] Hall, C. Big symplectic or orthogonal monodromy modulo ℓ. Duke Math. J. 141 (1) (2008), 179–203.Google Scholar
[IK04] Iwaniec, H. and Kowalski, E. Analytic number theory. Colloquium Publications (American Mathematical Society, 2004).Google Scholar
[Kat87] Katz, N. M. On the monodromy groups attached to certain families of exponential sums. Duke Math. J. 54 (1) (1987).Google Scholar
[Kat88] Katz, N. M. Gauss sums, Kloosterman sums, and monodromy Groups. Annals of Math. Stud. vol. 116 (Princeton University Press, 1988).Google Scholar
[Kat90] Katz, N. M. Exponential sums and differential equations. Annals of Math. Stud. vol. 124 (Princeton University Press, 1990).Google Scholar
[KS91] Katz, N. M. and Sarnak, P. Random matrices, Frobenius eigenvalues and monodromy. Colloquium Publications, vol 45 (American Mathematical Society, 1991).Google Scholar
[KS14] Kowalski, E. and Sawin, W. F. Kloosterman paths and the shape of exponential sums. Composit Math. 2014. To appear.Google Scholar
[Lam13] Lamzouri, Y. The distribution of short character sums. Math. Proc. Cam. Phil. Soc. 155 (2) (2013), 207–218.Google Scholar
[Lar90] Larsen, M. The normal distribution as a limit of generalised Sato-Tate measures. Unpublished note, http://mlarsen.math.indiana.edu/~larsen/papers/gauss.pdf (1990).Google Scholar
[LZ12] Lamzouri, Y. and Zaharescu, A. Randomness of character sums modulo m . J. Number Theory 132 (12) (2012), 2779–2792.Google Scholar
[Mac95] Macdonald, I. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Oxford University Press, second edition, 1995).Google Scholar
[Mic98] Michel, P. Minorations de sommes d'exponentielles. Duke Math. J. 95 (2) (1998).Google Scholar
[MZ11] Mak, K.-H. and Zaharescu, A. The distribution of values of short hybrid exponential sums on curves over finite fields. Math. Res. Lett. 18 (1) (2011), 155–174.CrossRefGoogle Scholar
[PG16] Perret-Gentil, C. Probabilistic aspects of short sums of trace functions over finite fields. PhD thesis (ETH Zürich, 2016).Google Scholar
[Pol14] Polymath, D. H. J. New equidistribution estimates of Zhang type. Algebra Number Theory 8 (9) (2014).Google Scholar
[Pro90] Proctor, R. A. A Schensted algorithm which models tensor representations of the orthogonal group. Canad. J. Math. 42 (1) (1990), 28–49.Google Scholar
[PV04] Pastur, L. and Vasilchuk, V. On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 (2004).Google Scholar
[Ram95] Ram, A. Characters of Brauer's centraliser algebras. Pacific J. Math. 169 (1) (1995).Google Scholar
[Reg81] Regev, A. Asymptotic values for degrees associated with strips of Young diagrams. Adv. Math. 41 (2) (1981), 115–136.Google Scholar
[Sag15] SageMath The Sage Mathematics Software System (Version 6.10), (2015). http://www.sagemath.org.Google Scholar
[Sel92] Selberg, A. Old and new conjectures and results about a class of Dirichlet series. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) University of Salerno (1992), pp. 367–385.Google Scholar
[Ser89] Serre, J.-P. Abelian ℓ-adic representations and elliptic curves. Research Notes in Mathematics, vol. 7 (Addison-Wesley, 1989).Google Scholar
[Sun86] Sundaram, S. On the combinatorics of representations of Sp(2n, ℂ). PhD thesis (Massachusetts Institute of Technology, 1986).Google Scholar
[Sun90] Sundaram, S. Orthogonal tableaux and an insertion algorithm for SO(2n + 1). J. Combin. Theory Ser. A 53 (2) (1990), 239–256.Google Scholar
[vdW34] van der Waerden, B. L. Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109 (1934), 13–16.CrossRefGoogle Scholar