Titlebook: Notes on Geometry and Arithmetic; Daniel Coray Textbook 2020 Springer Nature Switzerland AG 2020 algebraic varieties.rational points.cubic

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Diophantus of Alexandria,metic over the field of rational numbers. It was 1,300 years before Western mathematicians became interested in this type of problem (Bombelli, Viète, Bachet, Fermat), … on reading Diophantus to be precise. He also introduced new methods and a special symbol to express an unknown, which makes him an essential precursor of algebraic notation.

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Algebraic Closure; Affine Space,his is why they can quite easily be interpreted in terms of algebraic geometry, by adding if necessary rudiments of Galois theory. In this chapter, we introduce the algebraic and geometric concepts that seem best adapted to the arithmetic context.

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Projective Varieties; Conics and Quadrics,king in a projective setting. Arithmetic properties of projective varieties are strongly dependent on their geometry. The case of conics serves as a first illustration. Then we shall prove Springer’s and Brumer’s theorems on algebraic points on quadrics and intersections of quadrics.

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Euclidean Rings,al to be interested in Euclid’s division algorithm too, which has given rise to some impressive works. On formalizing the notion of the ., unexpected algorithms, revealed by new methods, have recently been discovered. There are also connections to several old unsolved conjectures.

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-Adic Completions,The field of .-adic numbers was introduced by Hensel at the beginning of the twentieth century. This remarkable idea greatly simplifies computations involving congruences, and is also of considerable theoretical interest, preparing the way for powerful generalizations.

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The Hasse Principle,The . asks the natural question: if a polynomial equation has non-trivial solutions in . and in .. for every prime ., can one deduce that it also has solutions in .? For quadratic forms, the answer is encouraging, but for more general situations this is only a “principle”, which may be verified or not.