Skip to main content Accessibility help
×
Home
Hostname: page-component-5d6d958fb5-27v8q Total loading time: 0.29 Render date: 2022-11-27T06:52:04.916Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Analytic number theory for 0-cycles

Published online by Cambridge University Press:  30 October 2017

WEIYAN CHEN*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA. e-mail: chen@math.uchicago.edu, wchen7@uchicago.edu

Abstract

There is a well-known analogy between integers and polynomials over 𝔽q, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorisation of effective 0-cycles on an arbitrary connected variety V over 𝔽q, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles are typically Poisson distributed.

Type
Research Article
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

[1] Arratia, R., Barbour, A. and Tavaré, S. On random polynomials over finite fields. Math. Proc. Cambridge Philos. Soc. 114 (1993), 347368.CrossRefGoogle Scholar
[2] Buchstab, A. Asymptotic estimation of a general number-theoretic function. Mat. Sb. (in Russian), 2 (44) (6) (1937), 12391246.Google Scholar
[3] Bender, E., Mashatan, A., Panario, D. and Richmond, B. Asymptotics of combinatorial structures with large smallest component. J. Combin. Theory Ser. A 107 (2004), 117125.CrossRefGoogle Scholar
[4] Chen, W. Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting. Preprint, arXiv:1603.03931 (2016).Google Scholar
[5] Church, T., Ellenberg, J. and Farb, B. Representation stability in cohomology and asymptotics for families of varieties over finite fields. Contemp. Math. 620 (2014), 154.CrossRefGoogle Scholar
[6] Deligne, P. La conjecture de Weil: II. Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.CrossRefGoogle Scholar
[7] Dickman, K. On the frequency of numbers containing prime factors of a certain relative magnitude. Arkiv főr Matematik, Astronomi och Fysik 22A (1930), (10) 114.Google Scholar
[8] Erdős, P. and Kac, M. The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62 (1940), 738742.CrossRefGoogle Scholar
[9] Flajolet, P. and Sedgewick, R.. Analytic Combinatorics (Cambridge University Press, Cambridge, 2009).CrossRefGoogle Scholar
[10] Flajolet, P. and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A 153 (1990), 165182.CrossRefGoogle Scholar
[11] Farb, B. and Wolfson, J. Étale homological stability and arithmetic statistics. Preprint arXiv:1512.00415 (2015).Google Scholar
[12] Granville, A. Prime divisors are Poisson distributed. Int. J. Number Theory 03 (01), (2011).Google Scholar
[13] Granville, A. The anatomy of integers and permutations. Preprint (2008).Google Scholar
[14] Granville, A. Cycle lengths in a permutation are typically Poisson distributed. Electron. J. Combin. 13 (2006).Google Scholar
[15] Hyde, T. and Lagarias, J. Polynomial splitting measures and cohomology of the pure braid group. Preprint, arXiv:1604.05359 (2016).Google Scholar
[16] Liu, Y. A generalisation of the Erdős–Kac theorem and its applications. Canad. Math. Bull. 47 (2004), 589606.CrossRefGoogle Scholar
[17] Lang, S. and Weil, A. Number of points of varieties in finite fields, in Amer. J. Math. 76 (1954), 819827. 2CrossRefGoogle Scholar
[18] Omar, M., Panario, D., Richmond, B. and Whitely, J. Asymptotics of largest components in combinatorial structures. Algorithmica 46 (3) (2006), 493503.CrossRefGoogle Scholar
[19] Poonen, B. Bertini theorems over finite fields. Ann. of Math. (2004), 1099–1127.Google Scholar
[20] Mumford, D. Abelian Varieties. Tata Inst. Fund. Research Stud. Math. (Oxford University Press, 1970).Google Scholar
[21] Ramaswami, V. On the number of positive integers less than x and free of prime divisors greater than x c. Bull. Amer. Math. Soc. 55 (12) (1949), 11221127.CrossRefGoogle Scholar
[22] Rhoades, R. Statistics of prime divisors in function fields. Int. J. Number Theory 05 (2009), 141.CrossRefGoogle Scholar
[23] Weil, A. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. 55 (1949), 497508.CrossRefGoogle Scholar
2
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.

Analytic number theory for 0-cycles
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.

Analytic number theory for 0-cycles
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.

Analytic number theory for 0-cycles
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? *