Ostrik V.V., Tsfasman M.A. Algebraic geometry and number theory: rational and elliptic curves

Translated from the Russian. — Moscow: MCCME, 2022. — 54 p. — ISBN 978-5-4439-4630-6.Many problems of number theory are beautifully solved used geometric methods, more precisely, by the methods of algebraic geometry - a branch of mathematics that studies curves, surfaces, etc., given by systems of polynomial equations. In the book, this is shown by the example of several beautiful problems in number theory related to the Pythagorean theorem.
Original edition: Острик В.В., Цфасман М.А. Алгебраическая геометрия и теория чисел: рациональные и эллиптические кривые /file/1232174/ (PDF), /file/322880/ (DJVU).Pythagorean triples.
Rational curves.
Elliptic curves.
Congruent numbers.
Addenda:
Proof of Theorem 1.
Fermat’s Last Theorem and Euler’s problem.
Pythagorean bricks.
How Diophantes solved arithmetical problems?
Answers, hints, solutions.