Joseph H. Silverman

John T. Tate

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.

Rational Points on Elliptic Curves

The arithmetic of elliptic curves

The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics)

Glenn Stevens

Gary Cornell

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.

Modular Forms and Fermat’s Last Theorem

Samuel G. Armistead

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1994.

Judeo-Spanish Ballads from Oral Tradition, II Carolingian Ballads, 1: Roncesvalles

Folk Literature of the Sephardic Jews, Vol. III

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.

Advanced Topics in the Arithmetic of Elliptic Curves

This book constitutes the thoroughly refereed post-proceedings of the International Conference on Cryptography and Lattices, CaLC 2001, held in Providence, RI, USA in March 2001. The 14 revised full papers presented together with an overview paper were carefully reviewed and selected for inclusion in the book. All current aspects of lattices and lattice reduction in cryptography, both for cryptographic construction and cryptographic analysis, are addressed.

International Conference, CaLC 2001, Providence, RI, USA, March 29-30, 2001. Revised Papers

Cryptography and Lattices

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of this book. Topics covered include the geometry and group structure of elliptic curves, the Nagell-Lutz theorem describing points of finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.

In New York City during the winter of 1922 and the spring of 1923, Mair Jose Benardete recorded the texts of the thirty-nine traditional ballads published in this volume. His collection, the beginning of Judeo-Spanish ballad research in America, was assembled when the oral tradition was still rich and vigorous among immigrants to New York from the Sephardic settlements of the Eastern Mediterranean and North Africa. Among the ballads are a number of rare text types, some never again recorded in the Sephardic communities of the United States, In addition, many of the texts provide new insights into the origins of the thematic traditions they represent. Samuel G. Armistead and Joseph H. Silverman have edited the ballads collected by Benardete, offering an English abstract and exhaustive bibliography for each ballad. In addition to placing each ballad within the context of its Sephardic variants, the bibliographies refer to the most important collections in the modern Castilian, Portuguese, Catalan, and Hispano-American traditions, to earlier (fifteenth- to seventeenth-century) evidence, and to any known analogs in other European traditions. The volume also includes a general bibliography, a thematic classification of the ballads, several indexes, and a glossary of exotic lexical elements. In an introduction, professors Armistead and Silverman present a documented survey of Judeo-Spanish ballad scholarship with particular attention to fieldwork in teh United States and elsewhere. Benardete himself attributed the decline of ballad singing among the Sephardim to a growing preference for phonographic recordings over traditional family singers. The need for further field-work increases as "Sephardic folkspeech and folklore retreat before the irresistible onslaught of the English language and modern American mass-media culture" (from the Introduction). This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1981.

Collected by Mair Jose Bernardete

Judeo-Spanish Ballads from New York

Josep María Sola-Solé

Studies on the History, Language, and Literature of the Jews in the Hispanic World

Hispania Judaica

Judeo-Spanish Ballads from Oral Tradition

Jeffrey Hoffstein

Jill Pipher

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.

An Introduction to Mathematical Cryptography

Marc Hindry

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

An Introduction

Diophantine Geometry

The Judeo-Spanish folk literature of the Sephardic Jews of Bosnia, and with it their uncommonly rich balladry, has remained largely unknown to Western scholars. Since their move to Sarajevo in the sixteenth century, Serob-Croatian has displaced their original Spanish, and the entire culture is rapidly approaching extinction. This book preserved for posterity three fundamentally important groups of these rare ballads: Kalmi Baruch's Spanski romanse; ballads collected from the readers of the Sarajevo newspaper Jevrejski Glas; and five previously unedited eighteenth-century Bosnian ballads from a manuscript in the Jewish National and University Library in Jerusalem. Notes, abstracts in English, reproductions of the music itself, and other scholarly aids serve to make this colorful and strangely modern literature fully accessible to Hispanists, folklorists, and all students of comparative literature and Judaic culture.

Joseph H. Silverman

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