Rational Points on Elliptic Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime

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Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime</title> <meta name="description" content="Rational Points on Elliptic Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime"> <meta name="keywords" content="Elliptic curves over finite fields, rational points."> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <meta name="citation_title" content="Rational Points on Elliptic Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime"> <meta name="citation_author" content="Gokhan Soydan"> <meta name="citation_author" content="Musa Demirci"> <meta name="citation_author" content="Nazli Yildiz Ikikardes"> <meta name="citation_author" content="Ismail Naci Cangul"> <meta name="citation_publication_date" content="2007/01/27"> <meta name="citation_journal_title" content="International Journal of Mathematical and Computational Sciences"> <meta name="citation_volume" content="1"> <meta 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href="https://publications.waset.org/search?q=Gokhan%20Soydan">Gokhan Soydan</a>, <a href="https://publications.waset.org/search?q=Musa%20Demirci"> Musa Demirci</a>, <a href="https://publications.waset.org/search?q=Nazli%20Yildiz%20Ikikardes"> Nazli Yildiz Ikikardes</a>, <a href="https://publications.waset.org/search?q=Ismail%20Naci%20Cangul"> Ismail Naci Cangul</a> </p> <p class="card-text"><strong>Abstract:</strong></p> <p>In this work, we consider the rational points on elliptic curves over finite fields Fp where p &equiv; 5 (mod 6). We obtain results on the number of points on an elliptic curve y2 &equiv; x3 + a3(mod p), where p &equiv; 5 (mod 6) is prime. We give some results concerning the sum of the abscissae of these points. A similar case where p &equiv; 1 (mod 6) is considered in [5]. The main difference between two cases is that when p &equiv; 5 (mod 6), all elements of Fp are cubic residues.</p> <iframe src="https://publications.waset.org/14639.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=Elliptic%20curves%20over%20finite%20fields" title="Elliptic curves over finite fields">Elliptic curves over finite fields</a>, <a href="https://publications.waset.org/search?q=rational%20points." title=" rational points."> rational points.</a> </p> <p class="card-text"><strong>Digital Object Identifier (DOI):</strong> <a href="https://doi.org/10.5281/zenodo.1335134" target="_blank">doi.org/10.5281/zenodo.1335134</a> </p> <a href="https://publications.waset.org/14639/rational-points-on-elliptic-curves-2-3-3y-x-a-inf-where-p-5mod-6-is-prime" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/14639/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/14639/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/14639/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/14639/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/14639/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/14639/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/14639/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/14639/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/14639/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/14639/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/14639.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">2250</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[1] Jones,G.A., Jones,J.M., Elementary Number Theory, Springer-Verlag,(1998),ISBN 3-540-76197-7<br>[2] Esmonde, J. & Murty, M. R., Problems in Algebraic Number Theory,Springer-Verlag, (1999), ISBN 0-387-98617-0.<br>[3] Schoof, R. , Counting points on elliptic curves over finite fields, Journalde Theorie des Nombres de Bordeaux, 7(1995), 219-254.<br>[4] Silverman, J.H., The Arithmetic of Elliptic Curves, Springer-Verlag,(1986), ISBN 0-387-96203-4.<br>[5] Demirci, M. & Soydan, G. & Cang├╝l, I. N., Rational points on theelliptic curves y2 = x3 + a3 (mod p) in Fpwhere p ≡ 1(mod 6) isprime, Rocky J.of Maths, ( to be printed ).<br>[6] Soydan, G. & Demirci, M. & Ikikarde┼ƒ, N. Y. & Cang├╝l, I. N., Rationalpoints on the elliptic curves y2 = x3 +a3 (mod p) in Fpwhere p≡ 5(mod 6) is prime, (submitted).<br>[7] Parshin, A. N., The Bogomolov-Miyaoka-Yau inequality for the arith-metical surfaces and its applications, Seminaire de Theorie des Nom-bres, Paris, 1986-87, 299-312, Progr. Math., 75, Birkhauser Boston, MA,1998.<br>[8] Kamienny, S., Some remarks on torsion in elliptic curves, Comm. Alg.23 (1995), no. 6, 2167-2169.<br>[9] Ono, K., Euler's concordant forms, Acta Arith. 78 (1996), no. 2, 101-123.<br>[10] Merel, L., Arithmetic of elliptic curves and Diophantine equations, Les XXemes Journees Arithmetiques (Limoges, 1997), J. Theor. Nombres Bordeaux 11 (1999), no. 1, 173-200.<br>[11] Serre, J.-P., Proprietes galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Account <li><a href="https://waset.org/profile">My Account</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Explore <li><a href="https://waset.org/disciplines">Disciplines</a></li> <li><a href="https://waset.org/conferences">Conferences</a></li> <li><a href="https://waset.org/conference-programs">Conference Program</a></li> <li><a href="https://waset.org/committees">Committees</a></li> <li><a href="https://publications.waset.org">Publications</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Research <li><a href="https://publications.waset.org/abstracts">Abstracts</a></li> <li><a href="https://publications.waset.org">Periodicals</a></li> <li><a href="https://publications.waset.org/archive">Archive</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Open Science <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Philosophy.pdf">Open Science Philosophy</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Award.pdf">Open Science Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Society-Open-Science-and-Open-Innovation.pdf">Open Innovation</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Postdoctoral-Fellowship-Award.pdf">Postdoctoral Fellowship Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Scholarly-Research-Review.pdf">Scholarly Research Review</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Support <li><a href="https://waset.org/page/support">Support</a></li> <li><a href="https://waset.org/profile/messages/create">Contact Us</a></li> <li><a href="https://waset.org/profile/messages/create">Report Abuse</a></li> </ul> </div> </div> </div> </div> </div> <div class="container text-center"> <hr style="margin-top:0;margin-bottom:.3rem;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" class="text-muted small">Creative Commons Attribution 4.0 International License</a> <div id="copy" class="mt-2">© 2024 World Academy of Science, Engineering and Technology</div> </div> </footer> <a href="javascript:" id="return-to-top"><i class="fas fa-arrow-up"></i></a> <div class="modal" id="modal-template"> <div class="modal-dialog"> <div class="modal-content"> <div class="row m-0 mt-1"> <div class="col-md-12"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> </div> </div> <div class="modal-body"></div> </div> </div> </div> <script src="https://cdn.waset.org/static/plugins/jquery-3.3.1.min.js"></script> <script src="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/js/bootstrap.bundle.min.js"></script> <script src="https://cdn.waset.org/static/js/site.js?v=150220211556"></script> <script> jQuery(document).ready(function() { /*jQuery.get("https://publications.waset.org/xhr/user-menu", function (response) { jQuery('#mainNavMenu').append(response); });*/ jQuery.get({ url: "https://publications.waset.org/xhr/user-menu", cache: false }).then(function(response){ jQuery('#mainNavMenu').append(response); }); }); </script> </body> </html>

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Rational Points on Elliptic Curves 2 3 3y = x + a inF , where p 5(mod 6) is Prime