000 02678nam a2200313Ia 4500
003 OSt
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020 _a9783319307572 (pbk.)
_c€ 49.99
040 _bENG
_cIISER-PR
041 _aENG
082 _a516.352
_bSIL
_223rd
100 _aSilverman, Joseph H.
_93698
222 _aMathematics
245 0 _aRational points on elliptic curves
250 _a2nd ed.
260 _aSwitzerland:
_bSpringer international Publishing,
_cc2015.
300 _axxii, 332p. :
_bill. ;
_c22cm.
440 _aUndergraduate texts in Mathematics
_93699
504 _aIncludes illutrations, list of notations, references and index.
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.
650 _aMathematics
_93700
650 _aGeometry and Arithmetic
_93701
650 _aPoints and Curves
_93702
700 _aTate, John T.
_93703
942 _cBK
_2ddc
947 _a4384.967831
948 _a0.22
999 _c3181
_d3181