000			02585nam a2200289Ia 4500
003			OSt
005			20240130183228.0
008			231229s9999 xx 000 0 und d
020			_a9783319307572
040			_cIISER BPR
041			_aEng
082			_a516.352 _bSIL _223rd
100			_aSilverman, Joseph H.
245		0	_aRational points on elliptic curves
250			_a2nd ed.
260			_aSwitzerland: _bSpringer international Publishing, _cc2015.
300			_axxii, 332p. : _bill (pbk). ; _c22cm.
440			_aUndergraduate texts in mathematics
520			_aThe theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume 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 Rational Points on Elliptic Curves. 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. Includes illutrations, list of notations, references and index.
650			_aMathematics
650			_aGeometry and Arithmetic
650			_aPoints and Curves
700			_aTate, John T.
942			_cBK _2ddc
947			_a4384.967831
948			_a0.22
999			_c3181 _d3181