國立虎尾科技大學 |

Rational Points on Elliptic Curves

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Rational Points on Elliptic Curves/ by Joseph H. Silverman, John T. Tate.
Author:	Silverman, Joseph H.
other author:	Tate, John T.
Description:	XXII, 332 p. 37 illus.online resource. :
Contained By:	Springer Nature eBook
Subject:	Algebraic geometry. -
Online resource:	https://doi.org/10.1007/978-3-319-18588-0
ISBN:	9783319185880

Rational Points on Elliptic Curves
Silverman, Joseph H.

Rational Points on Elliptic Curves[electronic resource] /by Joseph H. Silverman, John T. Tate. - 2nd ed. 2015. - XXII, 332 p. 37 illus.online resource. - Undergraduate Texts in Mathematics,0172-6056. - Undergraduate Texts in Mathematics,.

Introduction -- Geometry and Arithmetic -- Points of Finite Order -- The Group of Rational Points -- Cubic Curves over Finite Fields -- Integer Points on Cubic Curves -- Complex Multiplication.

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.

ISBN: 9783319185880

Standard No.: 10.1007/978-3-319-18588-0doiSubjects--Topical Terms:

1255324
Algebraic geometry.

LC Class. No.: QA564-609

Dewey Class. No.: 516.35

Rational Points on Elliptic Curves
LDR:03317nam a22004095i 4500 001 965509
003 DE-He213
005 20200705191843.0
007 cr nn 008mamaa
008 201211s2015 gw | s |||| 0|eng d
020 $a 9783319185880 $9 978-3-319-18588-0
024 7 $a 10.1007/978-3-319-18588-0 $2 doi
035 $a 978-3-319-18588-0
050 4 $a QA564-609
072 7 $a PBMW $2 bicssc
072 7 $a MAT012010 $2 bisacsh
072 7 $a PBMW $2 thema
082 0 4 $a 516.35 $2 23
100 1 $a Silverman, Joseph H. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 898181
245 1 0 $a Rational Points on Elliptic Curves $h [electronic resource] / $c by Joseph H. Silverman, John T. Tate.
250 $a 2nd ed. 2015.
264 1 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2015.
300 $a XXII, 332 p. 37 illus. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a Undergraduate Texts in Mathematics, $x 0172-6056
505 0 $a Introduction -- Geometry and Arithmetic -- Points of Finite Order -- The Group of Rational Points -- Cubic Curves over Finite Fields -- Integer Points on Cubic Curves -- Complex Multiplication.
520 $a 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.
650 0 $a Algebraic geometry. $3 1255324
650 0 $a Number theory. $3 527883
650 0 $a Data structures (Computer science). $3 680370
650 1 4 $a Algebraic Geometry. $3 670184
650 2 4 $a Number Theory. $3 672023
650 2 4 $a Data Structures and Information Theory. $3 1211601
700 1 $a Tate, John T. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1261102
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9783319185897
776 0 8 $i Printed edition: $z 9783319185873
776 0 8 $i Printed edition: $z 9783319307572
830 0 $a Undergraduate Texts in Mathematics, $x 0172-6056 $3 1253653
856 4 0 $u https://doi.org/10.1007/978-3-319-18588-0
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)