Elliptic curves : diophantine analysis
Material type:
TextLanguage: English Series: Grundlehren der mathematischen Wissenschaften ; 231Publication details: New York : Springer-Verlag, 1978 Description: xi, 261p. : ill. ; 25 cmISBN: 9783642057175Subject(s): Diophantine analysis | Curves, Elliptic | Algebra | Arithmetic | Curves | Diophantische Approximation | Diophantische Ungleichung | Elliptische Kurve | Function | TheoremDDC classification: 512.742 Online resources: Table of contents | Reviews Summary: It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.
| Item type | Current library | Call number | Status | Date due | Barcode |
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Book
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NISER LIBRARY | 512.742 LAN-E (Browse shelf(Opens below)) | Available | 17764 | |
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Book
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SMS Library | 512.742 LAN-E (Browse shelf(Opens below)) | Available | 25148 |
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| 512.74 SES-I Introduction to the theory of standard monomials | 512.74 SHO-E Exponential diophantine equations | 512.742 CLE-J Jacobi's lectures on dynamics | 512.742 LAN-E Elliptic curves : diophantine analysis | 512.742 LAN-I Introduction to algebraic and abelian functions | 512.742 MUR-I Introduction to abelian varieties | 512.743 BOR-L Linear algebraic groups |
Includes index.
Bibliography: p. [253]-259.
It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.
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