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Selmer Parity Of Quadratic Twists Of Elliptic Curves
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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 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 The Parity of Analytic Ranks Among Quadratic Twists of Elliptic Curves Over Number Fields by :
Download or read book The Parity of Analytic Ranks Among Quadratic Twists of Elliptic Curves Over Number Fields written by and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number eld and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in this family of elliptic curves.
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 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 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 Elliptic Curves and Related Topics by : H. Kisilevsky
Download or read book Elliptic Curves and Related Topics written by H. Kisilevsky and published by American Mathematical Soc.. This book was released on 1994-01-01 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.
Book Synopsis Elliptic Curves, Hilbert Modular Forms and Galois Deformations by : Laurent Berger
Download or read book Elliptic Curves, Hilbert Modular Forms and Galois Deformations written by Laurent Berger and published by Springer Science & Business Media. This book was released on 2013-06-13 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.
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 Elliptic Curves by : Dale Husemoller
Download or read book Elliptic Curves written by Dale Husemoller and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A general outline ofmuch ofthe material can be found in Tate's colloquium lectures reproduced as an article in Inven tiones [1974]. The first part grew out of Tate's 1961 Haverford Philips Lectures as an attempt to write something for publication c10sely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This inc1udes parts of the Introduction and the first six chapters The aim ofthis part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate teturned to Haverford to give again, in revised form, the originallectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan forthe book.
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 Variations on a Theme of Euler by : Takashi Ono
Download or read book Variations on a Theme of Euler written by Takashi Ono and published by Springer Science & Business Media. This book was released on 1994-11-30 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: The text, a translation of Dr. Ono's earlier work, provides a solution to this problem by employing three areas of mathematics: linear algebra, algebraic geometry, and simple algebras. This English-language edition presents a new chapter on arithmetic of quadratic maps, along with an appendix featuring a short survey of subsequent research on congruent numbers by Masanari Kida. The original appendix containing historical and scientific comments on Euler's Elements of Algebra is also included.
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 Introduction to Elliptic Curves and Modular Forms by : Neal I. Koblitz
Download or read book Introduction to Elliptic Curves and Modular Forms written by Neal I. Koblitz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.
Book Synopsis The Arithmetic of Elliptic Curves by : Joseph H. Silverman
Download or read book The Arithmetic of Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
Book Synopsis Advanced Topics in the Arithmetic of Elliptic Curves by : Joseph H. Silverman
Download or read book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 1994 with total page 546 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
