# eBook Rational Points on Elliptic Curves Undergraduate Texts in ´ ad325ddsc.merlotmotorsport.co.uk ´

[PDF / Epub] ❤ Rational Points on Elliptic Curves Undergraduate Texts in Mathematics ✅ John T Tate – Ad325ddsc.merlotmotorsport.co.uk The 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 opportunThe 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 euations and arithmetic geometryMost 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 uestion 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. The 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 euations and arithmetic geometryMost 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 uestion 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. 