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dc.contributor.authorDavid, Chantal
dc.contributor.authorJiménez Urroz, Jorge
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV
dc.date.accessioned2010-11-26T13:03:25Z
dc.date.available2010-11-26T13:03:25Z
dc.date.created2010-09
dc.date.issued2010-09
dc.identifier.citationDavid, C.; Jimenez, J. Square-free discriminants of Frobenius rings. "International journal of number theory", Setembre 2010, vol. 6, núm. 6, p. 1391-1412.
dc.identifier.issn1793-0421
dc.identifier.urihttp://hdl.handle.net/2117/10419
dc.description.abstractLet E be an elliptic curve over Q. It is well known that the ring of endomorphisms of $E_p$, the reduction of E modulo a prime p of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the Frobenius element $\pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a^2_p$−4p, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $\pi^{sf}_{E,r,h}(\chi)$ counting the number of primes p up to x such that $a^2_p$ is square-free and in the congruence class r modulo h. We give in this paper the precise asymptotic for $\pi^{sf}_{E,r,h}(\chi)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo p.
dc.format.extent22 p.
dc.language.isoeng
dc.rightsAttribution-NonCommercial-NoDerivs 3.0 Spain
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Ordres, reticles, estructures algebraiques ordenades
dc.subject.lcshCurves, Elliptic
dc.subject.lcshEndomorphism rings
dc.subject.lcshAlgebra, Abstract
dc.titleSquare-free discriminants of Frobenius rings
dc.typeArticle
dc.subject.lemacÀlgebra abstracta
dc.subject.lemacAnells (Àlgebra)
dc.subject.lemacCorbes el·líptiques
dc.contributor.groupUniversitat Politècnica de Catalunya. MAK - Matemàtica Aplicada a la Criptografia
dc.identifier.doi10.1142/S1793042110003599
dc.relation.publisherversionhttp://www.mathstat.concordia.ca/faculty/cdavid/PAPERS/SFdiscriminants-2008.pdf
dc.rights.accessOpen Access
local.identifier.drac3341862
dc.description.versionPostprint (published version)
local.citation.authorDavid, C.; Jimenez, J.
local.citation.publicationNameInternational journal of number theory
local.citation.volume6
local.citation.number6
local.citation.startingPage1391
local.citation.endingPage1412


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