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On the Average Exponent of Elliptic Curves Modulo p
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4734-5092
2014 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2014, no 8, 2265-2293 p.Article in journal (Refereed) Published
Abstract [en]

Given an elliptic curve E defined over <inline-graphic xlink:href="RNS280IM1" xmlns:xlink="http://www.w3.org/1999/xlink"/> and a prime p of good reduction, let <inline-graphic xlink:href="RNS280IM2" xmlns:xlink="http://www.w3.org/1999/xlink"/> denote the group of <inline-graphic xlink:href="RNS280IM3" xmlns:xlink="http://www.w3.org/1999/xlink"/>-points of the reduction of E modulo p, and let e(p) denote the exponent of this group. Assuming a certain form of the generalized Riemann hypothesis (GRH), we study the average of e(p) as <inline-graphic xlink:href="RNS280IM4" xmlns:xlink="http://www.w3.org/1999/xlink"/> ranges over primes of good reduction, and find that the average exponent essentially equals p center dot c(E), where the constant c(E)> 0 depends on E. For E without complex multiplication (CM), c(E) can be written as a rational number (depending on E) times a universal constant, <inline-graphic xlink:href="RNS280IM5" xmlns:xlink="http://www.w3.org/1999/xlink"/>, the product being over all primes q. Without assuming GRH, we can determine the average exponent when E has CM, as well as give an upper bound on the average in the non-CM case.

Place, publisher, year, edition, pages
2014. Vol. 2014, no 8, 2265-2293 p.
Keyword [en]
Artin Conjecture, Reductions, Cyclicity, Fields
National Category
Other Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-136366DOI: 10.1093/imrn/rns280ISI: 000334361600008Scopus ID: 2-s2.0-84896337544OAI: oai:DiVA.org:kth-136366DiVA: diva2:675923
Funder
Swedish Research Council
Note

QC 20140228

Available from: 2013-12-04 Created: 2013-12-04 Last updated: 2017-12-06Bibliographically approved

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Kurlberg, Pär

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