Author: Everett William HowePublisher:
ISBN:
Category :
Languages : en
Pages : 202
Book Description
Elliptic Curves and Ordinary Abelian Varieties Over Finite Fields
Author: Everett William HowePublisher:
ISBN:
Category :
Languages : en
Pages : 202
Book Description
Introduction to Abelian Varieties
Author: Vijaya Kumar MurtyPublisher: American Mathematical Soc.
ISBN: 0821811797
Category : Mathematics
Languages : en
Pages : 128
Book Description
This book presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory, and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.
Supersingular Abelian Varieties Over Finite Fields
Author: Hui ZhuModular Forms and Fermat’s Last Theorem
Author: Gary CornellPublisher: Springer Science & Business Media
ISBN: 1461219744
Category : Mathematics
Languages : en
Pages : 592
Book Description
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
Abelian Varieties and Decidability in Number Theory
Author: Caleb SpringerPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
This dissertation consists of two parts, both of which are focused upon problems and techniques from algebraic number theory and arithmetic geometry. In the first part, we consider abelian varieties defined over finite fields, which are a key object in cryptography in addition to being inherently intriguing in their own right. First, generalizing a theorem of Lenstra for elliptic curves, we present an explicit description of the group of rational points of a simple abelian variety over a finite field as a module over its endomorphism ring, under some technical conditions. Next, we present an algorithm for computing endomorphism rings in the case of ordinary abelian varieties of dimension 2, again under certain conditions, building on the work of Bisson and Sutherland. We prove the algorithm has subexponential running time by exploiting ideal class groups and class field theory. In the second part, we turn our attention to questions of decidability and definability for algebraic extensions of the rational numbers, in the vein of Hilbert's Tenth Problem and its generalizations. First, we show that a key technique for proving undecidability results fails for "most" subfields of the algebraic closure of the rational numbers. More specifically, we view the set of subfields of the algebraic closure of the rational numbers as a topological space, and prove there is a meager subset containing all subfields for which the ring of integers is existentially or universally definable. Finally, we present explicit families of infinite algebraic extensions of the rational numbers whose first-order theory is undecidable. This is achieved by leveraging the unit groups of totally imaginary quadratic extensions of totally real fields, building on the work of Martínez-Ranero, Utreras and Videla.
Abelian l-Adic Representations and Elliptic Curves
Author: Jean-Pierre SerrePublisher: CRC Press
ISBN: 1439863865
Category : Mathematics
Languages : en
Pages : 203
Book Description
This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one
Abelian Varieties with Small Isogeny Class and Applications to Cryptography
Author: Travis SchollPublisher:
ISBN:
Category :
Languages : en
Pages : 108
Book Description
An elliptic curve E over a finite field F[subscipt]q is called isolated if it admits few efficiently computable F[subscript]q-isogenies from E to a non-isomorphic curve. We present a variation on the CM method that constructs isolated curves. Assuming the Bateman-Horn conjecture, we show that there is negligible probability that a curve of cryptographic size constructed via this method is vulnerable to any known attack on the ECDLP. A special case of isolated curves is when the F[subscript]q-isogeny class contains only one F[subscript]q-isomorphism class. We call an elliptic curve, or abelian variety, super-isolated if it has this property. We give a simple characterization of super-isolated elliptic curves, and several examples of cryptographic size. We prove that there are only 2 super-isolated surfaces suitable for cryptographic use. Finally, we show that for any g[greater than or equal to]3, there are only finitely many super-isolated ordinary simple abelian varities of genus g. Essentially, we have an existence result in the practical range for genus g[less than or equal to]2, and a non-existence result for the impractical genera g[greater than or equal to]3.
Modular Curves and Abelian Varieties
Author: John CremonaPublisher: Springer Science & Business Media
ISBN: 9783764365868
Category : Mathematics
Languages : en
Pages : 308
Book Description
This book presents lectures from a conference on "Modular Curves and Abelian Varieties'' at the Centre de Recerca Matemàtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.
Rational Points on Elliptic Curves
Author: Joseph H. SilvermanPublisher: Springer Science & Business Media
ISBN: 1475742525
Category : Mathematics
Languages : en
Pages : 292
Book Description
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
The Arithmetic of Elliptic Curves
Author: Joseph H. SilvermanPublisher: Springer Science & Business Media
ISBN: 1475719205
Category : Mathematics
Languages : en
Pages : 414
Book Description
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.