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Selmer Ranks Of Quadratic Twists Of Elliptic Curves
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Book Synopsis Selmer Ranks of Quadratic Twists of Elliptic Curves by : Zev Klagsbrun
Download or read book Selmer Ranks of Quadratic Twists of Elliptic Curves written by Zev Klagsbrun and published by . This book was released on 2011 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis investigates the 2-Selmer rank in quadratic-twist families of elliptic curves defined over number fields, presenting new results in this area for curves having E(K)[2]=0 and E(K)[2] = Z/2Z. In particular, we show that all elliptic curves with E(K)[2]=0 have twists with 2-Selmer rank equal to r for every r & ge; 0 subject to the condition of constant 2-Selmer parity, and give a lower bound on the number of such twists as a function of the conductor. We do the same for all elliptic curves with E(K)[2] = Z/2Z that do not have a cyclic 4-isogeny defined over K(E[2]). Lastly, we present an infinite family of elliptic curves with coefficients in Q such that if 2 splits completely in K, then the 2-Selmer rank of EsuperF\super/K is bounded below by rsub2\sub(K) for every quadratic F/K.
Book Synopsis Selmer Parity of Quadratic Twists of Elliptic Curves by : Heng Su
Download or read book Selmer Parity of Quadratic Twists of Elliptic Curves written by Heng Su and published by . This book was released on 2015 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists EL by the norms of the finite part of conductors of quadratic extensions L/K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate's algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer's early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases.
Book Synopsis Rank 1 Quadratic Twists of Elliptic Curves and 3-Selmer Groups by : Zane Kun Li
Download or read book Rank 1 Quadratic Twists of Elliptic Curves and 3-Selmer Groups written by Zane Kun Li and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Selmer Ranks of Twists of Algebraic Curves by : Myungjun Yu
Download or read book Selmer Ranks of Twists of Algebraic Curves written by Myungjun Yu and published by . This book was released on 2016 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inspired by recent papers of Mazur-Rubin [8] and Klagsbrun-Mazur-Rubin [6], this thesis investigates Selmer ranks of twists of Jacobians of various algebraic curves over number field. For example, we find sufficient conditions on hyperelliptic curves C over a number field such that for any nonnegative integer r, there exist infinitely many quadratic twists of C whose Jacobians have 2-Selmer ranks equal to r. This theorem is even more generalized to the superelliptic curve case in this dissertation. We also present some results on 2-Selmer ranks of elliptic curves. In particular, we prove if the set of 2-Selmer ranks of quadratic twists of an elliptic curve over a number field contains an integer c, it contains all integers larger than c having the same parity as c.
Book Synopsis On the Average of P-selmer Rank in Quadratic Twist Families of Elliptic Curves Over Function Field by : Niudun Wang (Ph.D.)
Download or read book On the Average of P-selmer Rank in Quadratic Twist Families of Elliptic Curves Over Function Field written by Niudun Wang (Ph.D.) and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis focuses on studying the rank of Selmer groups in quadratic twist families of elliptic curves over function field. Some of the results are closely related to Poonen-Rains heuristics that hypothesizes the average of Selmer ranks of elliptic curves in general. We show in the first part that if the quadratic twist family of a given elliptic curve over Fq[t] with no Fq(t)-rational p-torsion points has an element whose Neron model has a multiplication reduction away from [infinity symbol], then the average [italicized p]-Selmer rank is [italicized p]+1 in large [italicized q]-limit for almost all primes [italicized p]. In the second part, we show that in the quadratic twist family of an elliptic curve over Fq[t] with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field, at least half of the quadratic twists have arbitrarily large 2-Selmer rank.
Book Synopsis On the Selmer Groups of Elliptic Curves in Quadratic Twist Families by : Siman Yat-Fai Wong
Download or read book On the Selmer Groups of Elliptic Curves in Quadratic Twist Families written by Siman Yat-Fai Wong and published by . This book was released on 1995 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Distribution of Selmer Groups of Quadratic Twists of a Family of Elliptic Curves by : Maosheng Xiong
Download or read book Distribution of Selmer Groups of Quadratic Twists of a Family of Elliptic Curves written by Maosheng Xiong and published by ProQuest. This book was released on 2007 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: As an most interesting example for the elliptic curve E n : y2 = x 3 - n2x, which is closely related with the congruent number problem, we study the distribution of the size of the six Selmer groups arising from the three 2-isogenies and their dual 2-isogenies. We also describe explicit formulas on the size of the six Selmer groups.
Book Synopsis Simultaneous Twists of Elliptic Curves and the Hasse Principle for Certain K3 Surfaces by :
Download or read book Simultaneous Twists of Elliptic Curves and the Hasse Principle for Certain K3 Surfaces written by and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.
Book Synopsis Ranks of Elliptic Curves and Random Matrix Theory by : J. B. Conrey
Download or read book Ranks of Elliptic Curves and Random Matrix Theory written by J. B. Conrey and published by Cambridge University Press. This book was released on 2007-02-08 with total page 5 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive volume introduces elliptic curves and the fundamentals of modeling by a family of random matrices.
Book Synopsis Elliptic Curves by : Susanne Schmitt
Download or read book Elliptic Curves written by Susanne Schmitt and published by Walter de Gruyter. This book was released on 2008-08-22 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.
Book Synopsis Selmer Groups and Ranks of Elliptic Curves by : Tengyao Wang
Download or read book Selmer Groups and Ranks of Elliptic Curves written by Tengyao Wang and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Kolyvagin Systems written by Barry Mazur and published by American Mathematical Soc.. This book was released on 2004 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their introduction by Kolyvagin, Euler systems have been used in several important applications in arithmetic algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's machinery is designed to provide an upper bound for the size of the Selmer group associated to the Cartier dual $T^*$. Given an Euler system, Kolyvagin produces a collection of cohomology classes which he calls ``derivative'' classes. It is these derivative classes which are used to bound the dual Selmer group. The starting point of the present memoir is the observation that Kolyvagin's systems of derivative classes satisfy stronger interrelations than have previously been recognized. We call a system of cohomology classes satisfying these stronger interrelations a Kolyvagin system. We show that the extra interrelations give Kolyvagin systems an interesting rigid structure which in many ways resembles (an enriched version of) the ``leading term'' of an $L$-function. By making use of the extra rigidity we also prove that Kolyvagin systems exist for many interesting representations for which no Euler system is known, and further that there are Kolyvagin systems for these representations which give rise to exact formulas for the size of the dual Selmer group, rather than just upper bounds.
Book Synopsis On the Selmer Group of Twists of Elliptic Curves with Q-rational Torsion Points by : Mathematical Sciences Research Institute (Berkeley, Calif.).
Download or read book On the Selmer Group of Twists of Elliptic Curves with Q-rational Torsion Points written by Mathematical Sciences Research Institute (Berkeley, Calif.). and published by . This book was released on 1987 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Families of Rank Zero Twists of Elliptic Curves by : Boris Iskra
Download or read book Families of Rank Zero Twists of Elliptic Curves written by Boris Iskra and published by . This book was released on 1998 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Issues in General and Specialized Mathematics Research: 2011 Edition by :
Download or read book Issues in General and Specialized Mathematics Research: 2011 Edition written by and published by ScholarlyEditions. This book was released on 2012-01-09 with total page 864 pages. Available in PDF, EPUB and Kindle. Book excerpt: Issues in General and Specialized Mathematics Research: 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about General and Specialized Mathematics Research. The editors have built Issues in General and Specialized Mathematics Research: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about General and Specialized Mathematics Research in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General and Specialized Mathematics Research: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.
Book Synopsis Ranks of Twists of Elliptic Curves by : Ron Fertig
Download or read book Ranks of Twists of Elliptic Curves written by Ron Fertig and published by . This book was released on 1999 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Lectures on the Mordell-Weil Theorem by : Jean-P. Serre
Download or read book Lectures on the Mordell-Weil Theorem written by Jean-P. Serre and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem. Comprehensive bibliographical references.
