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"Let be a prime. Let and be elliptic curves with isomorphic torsion modules and . Assume further that either every modules isomorphism admits a multiple with preserving the Weil pairing; or no isomorphism preserves the Weil pairing. This paper considers the problem of deciding if we are in case . Our approach is to consider the problem locally at a prime . Firstly, we determine the primes for which the local curves and contain enough information to decide between . Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of and , to decide between . We show that our results give a complete solution to the problem by local methods away from . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form and where is a prime; we also give incremental results on the Fermat equation . As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the torsion of two non-isogenous elliptic curves defined over "--

Curves, Elliptic. --- Number theory. --- Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Elliptic curves over global fields. --- Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Elliptic curves over local fields. --- Number theory -- Diophantine equations -- Higher degree equations; Fermat's equation.

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Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.

Arithmetical algebraic geometry --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Arithmetical algebraic geometry. --- Mathematics. --- Math --- Science

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There is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed, for instance, on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the André-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.

Arithmetical algebraic geometry. --- Logarithms. --- Logs (Logarithms) --- Algebra --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory

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E. Kähler: Geometria aritmetica.- L. Roth: Questioni di razionalità e varietà gruppali.- Seminars: B. Segre: Intorno alla geometria sopra un corpo di caratteristica p≠0 con particolare riguardo al caso p=2.

Algebra. --- Commutative algebra. --- Geometry. --- Geometry, Algebraic. --- Geometry, Analytic. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Arithmetical algebraic geometry --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Number theory --- Geometry, algebraic. --- Algebraic geometry

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Moduli theory is the study of how objects, typically in algebraic geometry but sometimes in other areas of mathematics, vary in families and is fundamental to an understanding of the objects themselves. First formalised in the 1960s, it represents a significant topic of modern mathematical research with strong connections to many areas of mathematics (including geometry, topology and number theory) and other disciplines such as theoretical physics. This book, which arose from a programme at the Isaac Newton Institute in Cambridge, is an ideal way for graduate students and more experienced researchers to become acquainted with the wealth of ideas and problems in moduli theory and related areas. The reader will find articles on both fundamental material and cutting-edge research topics, such as: algebraic stacks; BPS states and the P = W conjecture; stability conditions; derived differential geometry; and counting curves in algebraic varieties, all written by leading experts.

Moduli theory. --- Arithmetical algebraic geometry. --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Geometry, Algebraic. --- Moduli theory --- Algebraic geometry --- Geometry