Skip to main content Accessibility help

Gaussian distribution of short sums of trace functions over finite fields



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.



Hide All
[BRR86] Bhattacharya, R. N. and Ranga Rao, R. Normal Approximation and Asymptotic Expansions (Robert E. Krieger Publishing Co., 1986). Reprint of the 1976 original.
[DE52] Davenport, H. and Erdős, P. The distribution of quadratic and higher residues. Publ. Math. Debrecen 2 (1952), 252265.
[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).
[Del80] Deligne, P. La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. 52 (1) (1980), 137252.
[DS94] Diaconis, P. and Shahshahani, M. On the eigenvalues of random matrices. J. Appl. Probab. 31 (1994), 4962.
[FH91] Fulton, W. and Harris, J. Representation theory. Graduate Texts in Mathematics, vol. 129 (Springer, 1991).
[FKM14a] Fouvry, É., Kowalski, E. and Michel, P. Trace functions over finite fields and applications. (December 2014).
[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. 735.
[FKM15a] Fouvry, É., Kowalski, E. and Michel, P. Algebraic twists of modular forms and Hecke orbits. Geom. Funct. Anal. 25 (2) (2015), 580657.
[FKM15b] Fouvry, É., Kowalski, E. and Michel, P. A study in sums of products. Philos. Trans. A 373 (2040) (2015).
[FM02] Fouvry, É. and Michel, P. A la recherche de petites sommes d'exponentielles. Ann. Inst. Fourier (Grenoble) 52 (1) (2002), 4780.
[FM03] Fouvry, É. and Michel, P. Sommes de modules de sommes d'exponentielles. Pacific J. Math. 209 (2) (2003).
[Gut05] Gut, A. Probability: a Graduate Course. Springer texts in statistics (Springer, 2005).
[Hal08] Hall, C. Big symplectic or orthogonal monodromy modulo ℓ. Duke Math. J. 141 (1) (2008), 179203.
[IK04] Iwaniec, H. and Kowalski, E. Analytic number theory. Colloquium Publications (American Mathematical Society, 2004).
[Kat87] Katz, N. M. On the monodromy groups attached to certain families of exponential sums. Duke Math. J. 54 (1) (1987).
[Kat88] Katz, N. M. Gauss sums, Kloosterman sums, and monodromy Groups. Annals of Math. Stud. vol. 116 (Princeton University Press, 1988).
[Kat90] Katz, N. M. Exponential sums and differential equations. Annals of Math. Stud. vol. 124 (Princeton University Press, 1990).
[KS91] Katz, N. M. and Sarnak, P. Random matrices, Frobenius eigenvalues and monodromy. Colloquium Publications, vol 45 (American Mathematical Society, 1991).
[KS14] Kowalski, E. and Sawin, W. F. Kloosterman paths and the shape of exponential sums. Composit Math. 2014. To appear.
[Lam13] Lamzouri, Y. The distribution of short character sums. Math. Proc. Cam. Phil. Soc. 155 (2) (2013), 207218.
[Lar90] Larsen, M. The normal distribution as a limit of generalised Sato-Tate measures. Unpublished note, (1990).
[LZ12] Lamzouri, Y. and Zaharescu, A. Randomness of character sums modulo m . J. Number Theory 132 (12) (2012), 27792792.
[Mac95] Macdonald, I. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Oxford University Press, second edition, 1995).
[Mic98] Michel, P. Minorations de sommes d'exponentielles. Duke Math. J. 95 (2) (1998).
[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), 155174.
[PG16] Perret-Gentil, C. Probabilistic aspects of short sums of trace functions over finite fields. PhD thesis (ETH Zürich, 2016).
[Pol14] Polymath, D. H. J. New equidistribution estimates of Zhang type. Algebra Number Theory 8 (9) (2014).
[Pro90] Proctor, R. A. A Schensted algorithm which models tensor representations of the orthogonal group. Canad. J. Math. 42 (1) (1990), 2849.
[PV04] Pastur, L. and Vasilchuk, V. On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 (2004).
[Ram95] Ram, A. Characters of Brauer's centraliser algebras. Pacific J. Math. 169 (1) (1995).
[Reg81] Regev, A. Asymptotic values for degrees associated with strips of Young diagrams. Adv. Math. 41 (2) (1981), 115136.
[Sag15] SageMath The Sage Mathematics Software System (Version 6.10), (2015).
[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. 367385.
[Ser89] Serre, J.-P. Abelian ℓ-adic representations and elliptic curves. Research Notes in Mathematics, vol. 7 (Addison-Wesley, 1989).
[Sun86] Sundaram, S. On the combinatorics of representations of Sp(2n, ℂ). PhD thesis (Massachusetts Institute of Technology, 1986).
[Sun90] Sundaram, S. Orthogonal tableaux and an insertion algorithm for SO(2n + 1). J. Combin. Theory Ser. A 53 (2) (1990), 239256.
[vdW34] van der Waerden, B. L. Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109 (1934), 1316.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed