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Faltings's Theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field. Background Let ''C'' be a non-singular algebraic curve of genus ''g'' over Q. Then the set of rational points on ''C'' may be determined as follows: * Case ''g'' = 0: no points or infinitely many; ''C'' is handled as a conic section. * Case ''g'' = 1: no points, or ''C'' is an elliptic curve and its rational points form a finitely generated abelian group (''Mordell's Theorem'', later generalized to the Mordell–Weil theorem). Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. * Case ''g'' > 1: according to the Mordell conjecture, now Faltings's theorem, ''C'' has only a finite number of rational poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mazur's Torsion Theorem
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order (group theory), order of the torsion group of an abelian variety over a number field can be bounded in terms of the Dimension of an algebraic variety, dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of elliptic curves. Elliptic curves From 1906 to 1911, Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals. He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups: * ''C''''n'' with 1 ≤ ''n'' ≤ 10, where ''C''''n'' denotes the cyclic group of order ''n''; * ''C''12; * ''C''2n × ''C''2 with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. Bombieri won the Fields Medal in 1974 for his contributions to large sieve mathematics, conceptualized by Linnick 1941, and its application to the distribution of prime numbers. Career Bombieri published his first mathematical paper in 1957 when he was 16 years old. In 1963 at age 22 he earned his first degree (Laurea) in mathematics from the Università degli Studi di Milano under the supervision of Giovanni Ricci and then studied at Trinity College, Cambridge with Harold Davenport. Bombieri was an assistant professor (1963–1965) and then a full professor (1965–1966) at the Università di Cagliari, at the Università di Pisa in 1966–1974, and then at the Sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
