## Arithmetic of Algebraic CurvesAuthor S.A. Stepanov thoroughly investigates the current state of the theory of Diophantine equations and its related methods. Discussions focus on arithmetic, algebraic-geometric, and logical aspects of the problem. Designed for students as well as researchers, the book includes over 250 excercises accompanied by hints, instructions, and references. Written in a clear manner, this text does not require readers to have special knowledge of modern methods of algebraic geometry. |

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Abelian group absolute value algebraic closure algebraic curve algebraic numbers algebraic set arbitrary Archimedean automorphism coefficients congruence Consequently coordinates coprime Corollary defined definition denote differential Diophantine equations Div(F equivalent exists field F field K field k0 field of algebraic field of rational finite extension finite field finite number following statements function f functional prime divisor genus g hence Hint holds true homomorphism inequality infinite integral solutions irreducible polynomial isomorphic Lemma Let us consider linear linearly local ring modulo multiplicative n-tuples non-Archimedean nonnegative nonstandard nontrivial nonzero element number of solutions numbers Q obtain p-adic parameter polynomial f positive integer prime divisor prime ideal prime number principal divisor Proof Prove the validity quadratic quadratic nonresidue quotient group rational functions rational numbers rational points real numbers relation result Riemann-Roch theorem ring Fp[x satisfies the condition solvable space subset valuation variety wp(A zero

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