Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-2qt69 Total loading time: 0.311 Render date: 2022-08-12T01:57:19.646Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

An inverse theorem for Gowers norms of trace functions over Fp

Published online by Cambridge University Press:  25 April 2013

ÉTIENNE FOUVRY
Affiliation:
Université Paris Sud, Laboratoire de Mathématique, Campus d'Orsay, 91405 Orsay Cedex, France. e-mail: etienne.fouvry@math.u-psud.fr
EMMANUEL KOWALSKI
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland. e-mail: kowalski@math.ethz.ch
PHILIPPE MICHEL
Affiliation:
EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland. e-mail: philippe.michel@epfl.ch

Abstract

We study the Gowers uniformity norms of functions over Z/pZ which are trace functions of ℓ-adic sheaves. On the one hand, we establish a strong inverse theorem for these functions, and on the other hand this gives many explicit examples of functions with Gowers norms of size comparable to that of “random” functions.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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

[1]Chatzidakis, Z., van den Dries, L. and Macintyre, A.Definable sets over finite fields. J. Reine Angew. Math. 427 (1992), 107135.Google Scholar
[2]Deligne, P.Cohomologie étale, S.G.A 4½. Lecture Notes in Math. 569 (Springer Verlag, 1977).CrossRefGoogle Scholar
[3]Deligne, P.La conjecture de Weil, II. Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137252.CrossRefGoogle Scholar
[4]Esnault, H. and Kerz, M.A finiteness theorem for Galois representations of function fields over finite fields (after Deligne). Acta Math. Vietnam. 37 (2012), 351–362; arXiv:1208.0128v3.Google Scholar
[5]Fouvry, É., Kowalski, E. and Michel, Ph. Algebraic twists of modular forms and Hecke orbits. preprint (2012); arXiv:1210.0617v4.Google Scholar
[6]Fouvry, É., Kowalski, E. and Michel, Ph. Counting sheaves using spherical codes, preprint (2012); arXiv:1210.0851v2.Google Scholar
[7]Fouvry, É., Kowalski, E. and Michel, Ph. Algebraic trace functions over the primes, preprint (2012); arXiv:1211.6043v1.Google Scholar
[8]Fouvry, É., Kowalski, E. and Michel, Ph. Trace norms over finite fields, in preparation.Google Scholar
[9]Fouvry, É., Michel, P., Rivat, J. and Sárközy, A.On the pseudorandomness of the signs of Kloosterman sums. J. Australian Math. Soc. 77 (December 2004), 425436.CrossRefGoogle Scholar
[10]Green, B. J. and Tao, T.The distribution of polynomials over finite fields, with applications to the Gowers norms. Contrib. Discr. Math. 4 (2009), 136; arXiv:0711.3191.Google Scholar
[11]Green, B. J., Tao, T. and Ziegler, T.An inverse theorem for the Gowers Us+1[N]-norm. Annals of Math. 176 (2012), 12311372; arXiv:1009.3998.CrossRefGoogle Scholar
[12]Katz, N. M.Gauss sums, Kloosterman sums and monodromy groups. Annals of Math. Studies 116 (Princeton University Press, 1988).Google Scholar
[13]Katz, N. M.Exponential sums and differential equations. Annals of Math. Studies 124 (Princeton University Press, 1990).Google Scholar
[14]Katz, N. M.Moments, monodromy and perversity. Annals of Math. Studies 159 (Princeton University Press, 2005).Google Scholar
[15]Katz, N. M.Convolution and equidistribution: Sato–Tate theorems for finite-field Mellin transforms. Annals of Math. Studies 180 (Princeton University Press, 2011).Google Scholar
[16]Liu, H.Gowers uniformity norm and pseudorandom measures of the pseudorandom binary sequences. Internat. J. Number Theory 7 (2005), 12791302.CrossRefGoogle Scholar
[17]Niederreiter, H. and Rivat, J.On the Gowers norms of pseudorandom binary sequences. Bull. Aust. Math. Soc. 79 (2009), 259271.CrossRefGoogle Scholar
[18]Tao, T. and Vu, V.Additive Combinatorics. Cambridge Studies in Advanced Math. 105 (Cambridge University Press, 2006).Google Scholar
[19]Tao, T. and Ziegler, T.The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Anal. PDE 3 (2010), 120; arXiv:0810.5527.CrossRefGoogle Scholar
[20]Tao, T. and Ziegler, T.The inverse conjecture for the Gowers norm over finite fields in low characteristics. Ann. Comb. 16 (2012), 121188; arXiv:1101.1469.CrossRefGoogle Scholar
4
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

An inverse theorem for Gowers norms of trace functions over Fp
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

An inverse theorem for Gowers norms of trace functions over Fp
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

An inverse theorem for Gowers norms of trace functions over Fp
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *