Elliptic Diophantine Equations : A Concrete Approach Via the Elliptic Logarithm.

By: Tzanakis, NikosSeries: De Gruyter Series in Discrete Mathematics and Applications SerPublisher: Berlin/Boston : De Gruyter, Inc., 2013Copyright date: ©2013Description: 1 online resource (196 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783110281149Subject(s): Diophantine equations.;Elliptic functionsGenre/Form: Electronic books. Additional physical formats: Print version:: Elliptic Diophantine Equations : A Concrete Approach Via the Elliptic LogarithmDDC classification: 512.72 LOC classification: QA242.T93 2013ebOnline resources: Click to View
Contents:
Intro -- Preface -- 1 Elliptic curves and equations -- 1.1 A general overview -- 1.2 Elliptic curves and the Mordell-Weil Theorem -- 2 Heights -- 2.1 Notations and facts -- 2.2 Absolute values in a number field -- 2.3 Heights: Absolute and logarithmic -- 2.4 A formula for the absolute logarithmic height -- 2.5 Heights of points on an elliptic curve -- 2.6 The canonical height -- 3 Weierstrass equations over ℂ and ℝ -- 3.1 The Weierstrass P function -- 3.2 The Weierstrass equation -- 3.3 ψ : E(ℂ) → ℂ /Λ -- 3.4 Weierstrass equations with real coefficients -- 3.4.1 Δ > 0 -- 3.4.2 Δ < 0 -- 3.4.3 Explicit expressions for the periods -- 3.4.4 Computing ω1 and ω2 in practice -- 3.5 ψ : E(ℝ) → ℂ /Λ and ɽ : E(ℝ)→ ℝ/ℤω1 -- 4 The elliptic logarithm method -- 5 Linear form for theWeierstrass equation -- 6 Linear form for the quartic equation -- 7 Linear form for simultaneous Pell equations -- 8 Linear form for the general elliptic equation -- 8.1 A short Weierstrassmodel -- 8.2 Puiseux series -- 8.3 Large solutions -- 8.4 The elliptic integrals -- 8.5 Computing in practice B1 of Proposition 8.3.2 -- 8.6 Computing in practice B2 and c9 of Proposition 8.4.2 -- 8.7 The linear form L(P) / and its upper bound -- 9 Bound for the coefficients of the linear form -- 9.1 Lower bound for linear forms in elliptic logarithms -- 9.2 Computational remarks -- 9.3 Weierstrass equation example -- 9.4 Quartic equation example -- 9.5 Simultaneous Pell equations example -- 9.6 General elliptic equation: A quintic example -- 10 Reducing the bound obtained in Chapter 9 -- 10.1 Reduction using the LLL-algorithm -- 10.2 Examples -- 10.2.1 Weierstrass equation -- 10.2.2 Quartic equation -- 10.2.3 System of simultaneous Pell equations -- 10.2.4 General elliptic equation: A quintic example -- 11 S-integer solutions ofWeierstrass equations.
11.1 The formal group of C and p-adic elliptic logarithms -- 11.2 Points with coordinates in ℤS -- 11.3 The p-adic reduction -- 11.4 Example -- List of symbols -- Bibliography -- Index.
Summary: This series is devoted to the publication of high-level monographs which cover the whole spectrum of current discrete mathematics and its applications in various fields. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of discrete mathematics.
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Intro -- Preface -- 1 Elliptic curves and equations -- 1.1 A general overview -- 1.2 Elliptic curves and the Mordell-Weil Theorem -- 2 Heights -- 2.1 Notations and facts -- 2.2 Absolute values in a number field -- 2.3 Heights: Absolute and logarithmic -- 2.4 A formula for the absolute logarithmic height -- 2.5 Heights of points on an elliptic curve -- 2.6 The canonical height -- 3 Weierstrass equations over ℂ and ℝ -- 3.1 The Weierstrass P function -- 3.2 The Weierstrass equation -- 3.3 ψ : E(ℂ) → ℂ /Λ -- 3.4 Weierstrass equations with real coefficients -- 3.4.1 Δ > 0 -- 3.4.2 Δ < 0 -- 3.4.3 Explicit expressions for the periods -- 3.4.4 Computing ω1 and ω2 in practice -- 3.5 ψ : E(ℝ) → ℂ /Λ and ɽ : E(ℝ)→ ℝ/ℤω1 -- 4 The elliptic logarithm method -- 5 Linear form for theWeierstrass equation -- 6 Linear form for the quartic equation -- 7 Linear form for simultaneous Pell equations -- 8 Linear form for the general elliptic equation -- 8.1 A short Weierstrassmodel -- 8.2 Puiseux series -- 8.3 Large solutions -- 8.4 The elliptic integrals -- 8.5 Computing in practice B1 of Proposition 8.3.2 -- 8.6 Computing in practice B2 and c9 of Proposition 8.4.2 -- 8.7 The linear form L(P) / and its upper bound -- 9 Bound for the coefficients of the linear form -- 9.1 Lower bound for linear forms in elliptic logarithms -- 9.2 Computational remarks -- 9.3 Weierstrass equation example -- 9.4 Quartic equation example -- 9.5 Simultaneous Pell equations example -- 9.6 General elliptic equation: A quintic example -- 10 Reducing the bound obtained in Chapter 9 -- 10.1 Reduction using the LLL-algorithm -- 10.2 Examples -- 10.2.1 Weierstrass equation -- 10.2.2 Quartic equation -- 10.2.3 System of simultaneous Pell equations -- 10.2.4 General elliptic equation: A quintic example -- 11 S-integer solutions ofWeierstrass equations.

11.1 The formal group of C and p-adic elliptic logarithms -- 11.2 Points with coordinates in ℤS -- 11.3 The p-adic reduction -- 11.4 Example -- List of symbols -- Bibliography -- Index.

This series is devoted to the publication of high-level monographs which cover the whole spectrum of current discrete mathematics and its applications in various fields. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of discrete mathematics.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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