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Mordell's Conjecture
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field \mathbb of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing \mathbb by any number field. Background Let C be a non-singular algebraic curve of genus g over \mathbb. Then the set of rational points on C may be determined as follows: * When g=0, there are either no points or infinitely many. In such cases, C may be handled as a conic section. * When g=1, if there are any points, then C is an elliptic curve and its rational points form a finitely generated abelian group. (This is ''Mordell's Theorem'', later generalized to the Mordell–Weil theorem.) Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. * When g>1, according to Faltings's theorem, C has only ... [...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 fie ... [...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 of the torsion group of an abelian variety over a number field can be bounded in terms of the 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 1 ≤ ''n'' ≤ 4, where × denotes the direct sum. At ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''p''/''q'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''p''/''q'' and ''α'' may not decrease if ''p''/''q'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation. Contributions In formulating Vojta's conjecture, he pointed out the possible existence of parallels between the Nevanlinna theory of complex analysis, and diophantine analysis in the circle of ideas around the Mordell conjecture and abc conjecture. This suggested the importance of the ''integer solutions'' (affine space) aspect of diophantine equations. Vojta wrote the .dvi-previewer xdvi. He also wrote a vi clone. Education and career He was an undergraduate student at the University of Minnesota, where he became a Putnam Fellow in 1977, and a doctoral student at Harvard University (1983). He currently is a professor in the Department of Mathematics at the University of California, Berkeley. Awards and honors In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Modular Variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943. Siegel modular varieties are the most basic examples of Shimura varieties. Siegel modular varieties generalize moduli spaces of elliptic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions. They also have applications to black hole entropy and conformal field theory. Construction The Siegel modular variety ''A''''g'', which parametrize principally polarized abelian varieties of dimension ''g'', can be constructed as the complex analytic spaces constructed as the quotient of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the ''classical'' or ''naive height'' over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale. Significance Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Néron Model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), in other words the "best possible" group scheme ''AR'' defined over ''R'' corresponding to ''AK''. They were introduced by for abelian varieties over the quotient field of a Dedekind domain ''R'' with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains. Definition Suppose that ''R'' is a Dedekind domain with field of fractions ''K'', and suppose that ''AK'' is a smooth separated scheme over ''K'' (such as an abelian variety). Then a Néron model of ''AK'' is defined to be a smooth morphism, smooth Separated morphism, separated scheme ''AR'' over ''R'' with fiber ''AK'' that is universal in the following sense. :If ''X'' is a smooth separated scheme over ''R'' then any ''K''-mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture. Statement of the conjecture Let ''V'' be a smooth projective variety over a field ''k'' which is finitely generated over its prime field. Let ''k''s be a separable closure of ''k'', and let ''G'' be the absolute Galois group Gal(''k''s/''k'') of ''k''. Fix a prime number ℓ which is invertible in ''k''. Consider the ℓ-adic cohomology groups (coefficients in the ℓ-adic integers Zℓ, scalars then extended to the ℓ-adic numbers Qℓ) of the base extension of ''V'' to ''k''s; these groups are representations of ''G''. For any ''i'' ≥ 0, a codimension-''i'' subvariety of ''V'' (understood to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksei Parshin
Aleksei Nikolaevich Parshin (; 7 November 1942 – 18 June 2022) was a Russian mathematician, specializing in arithmetic geometry. He is most well-known for his role in the proof of the Mordell conjecture. Education and career Parshin entered the Faculty of Mathematics and Mechanics of Moscow State University in 1959 and graduated in 1964. He then enrolled as a graduate student at the Steklov Institute of Mathematics, where he received his '' Kand. Nauk'' (Ph.D.) in 1968 under Igor Shafarevich. In 1983, he received his '' Doctor Nauk'' (doctorate of sciences) from Moscow State University. Parshin became a junior research fellow at the Steklov Institute of Mathematics in Moscow in 1968, later becoming a senior and leading research fellow. He became the head of its Department of Algebra in 1995. He also taught at Moscow State University. Research In his 1968 thesis, Parshin proved that the Mordell conjecture is a logical consequence of Shafarevich's finiteness conjecture conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Place (mathematics)
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the sum of their squares, equal two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Reduction
In most contexts, the concept of good denotes the conduct that should be preferred when posed with a choice between possible actions. Good is generally considered to be the opposite of evil. The specific meaning and etymology of the term and its associated translations among ancient and contemporary languages show substantial variation in its inflection and meaning, depending on circumstances of place and history, or of philosophical or religious context. History of Western ideas Every language has a word expressing ''good'' in the sense of "having the right or desirable quality" (ἀρετή) and ''bad'' in the sense "undesirable". A sense of moral judgment and a distinction "right and wrong, good and bad" are cultural universals. Plato and Aristotle Although the history of the origin of the use of the concept and meaning of "good" are diverse, the notable discussions of Plato and Aristotle on this subject have been of significant historical effect. The first references ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
