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Tate–Shafarevich Group
In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K^/K) is the absolute Galois group of , that become trivial in all of the completions of (i.e., the real and complex completions as well as the -adic fields obtained from by completing with respect to all its Archimedean and non Archimedean valuations ). Thus, in terms of Galois cohomology, can be defined as :\bigcap_v\mathrm\left(H^1\left(G_K,A\right)\rightarrow H^1\left(G_,A_v\right)\right). This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation , where is the Cyrillic letter " Sha", for Shafarevich, replacing the older notation or . Elements of the Tate–Shafarevich group Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homog ... [...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] |
Genus Of An Algebraic Curve
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic \chi, via the relationship \chi=2-2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads \chi=2-2g-b. In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William A
William is a masculine given name of Germanic origin. It became popular in England after the Norman conquest in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will or Wil, Wills, Willy, Willie, Bill, Billie, and Billy. A common Irish form is Liam. Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie). Female forms include Willa, Willemina, Wilma and Wilhelmina. Etymology William is related to the German given name ''Wilhelm''. Both ultimately descend from Proto-Germanic ''*Wiljahelmaz'', with a direct cognate also in the Old Norse name ''Vilhjalmr'' and a West Germanic borrowing into Medieval Latin ''Willelmus''. The Proto-Germanic name is a compound of *''wiljô'' "will, wish, desire" and *''helmaz'' "helm, helmet".Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Swinnerton-Dyer
Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet, (2 August 1927 – 26 December 2018) was an English mathematician specialising in number theory at the University of Cambridge. As a mathematician he was best known for his part in the Birch and Swinnerton-Dyer conjecture relating algebraic properties of elliptic curves to special values of L-functions, which was developed with Bryan Birch during the first half of the 1960s with the help of machine computation, and for his work on the Titan operating system. Biography Swinnerton-Dyer was the son of Sir Leonard Schroeder Swinnerton Dyer, 15th Baronet, and his wife Barbara, daughter of Hereward Brackenbury. He was educated at the Dragon School in Oxford, Eton College and Trinity College, Cambridge, where he was supervised by J. E. Littlewood, and spent a year abroad as a Commonwealth Fund Fellow at the University of Chicago. He was later made a Fellow of Trinity, and was Master of St Catharine's College from 1973 to 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local field k, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology: :\displaystyle H^r(k,M)\times H^(k,M')\rightarrow H^2(k,\mathbb_m)=\Q/ \Z where M is a finite group scheme, M' its dual \operatorname(M,G_m), and \mathbb_m is the multiplicative group. For a local field of characteristic p>0, the statement is similar, except that the pairing takes values in H^2(k, \mu) = \bigcup_ \tfrac \Z/\Z. The statement also holds when k is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case. Global Tate duality Given a finite group scheme M over a global field k, global Tate duality relates the cohomology of M with that of M' = \operatorname(M,G_m) using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Pairing
In mathematics, a pairing is an ''R''-Bilinear map#Modules, bilinear map from the Cartesian product of two ''R''-Module (mathematics), modules, where the underlying Ring (mathematics), ring ''R'' is Commutative ring, commutative. Definition Let ''R'' be a commutative ring with Unit (mathematics), unit, and let ''M'', ''N'' and ''L'' be Module (mathematics), ''R''-modules. A pairing is any ''R''-bilinear map e:M \times N \to L. That is, it satisfies :e(r\cdot m,n)=e(m,r \cdot n)=r\cdot e(m,n), :e(m_1+m_2,n)=e(m_1,n)+e(m_2,n) and e(m,n_1+n_2)=e(m,n_1)+e(m,n_2) for any r \in R and any m,m_1,m_2 \in M and any n,n_1,n_2 \in N . Equivalently, a pairing is an ''R''-linear map :M \otimes_R N \to L where M \otimes_R N denotes the Tensor product of modules, tensor product of ''M'' and ''N''. A pairing can also be considered as an module homomorphism, ''R''-linear map \Phi : M \to \operatorname_ (N, L) , which matches the first definition by setting \Phi (m) (n) := e(m,n) . A pairin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Abelian Group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a ''generating set'' of G or that x_1,\dots, x_s ''generate'' G. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent that divides its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich (see '' Golod–Shafarevich theorem''), and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modularity Theorem
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the Taniyama–Shimura conjecture, Taniyama–Shimura–Weil conjecture, or the modularity conjecture for elliptic curves. Statement The theorem states that any elliptic curve over \Q can be obtained via a rational map with integer coefficients from the classical modular curve for some integer ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Kolyvagin
Victor Alexandrovich Kolyvagin (, born 11 March, 1955) is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Last Theorem. Career Kolyvagin received his Ph.D. in Mathematics in 1981 from Moscow State University, where his advisor was Yuri I. Manin. He then worked at Steklov Institute of Mathematics in Moscow until 1994. Since 1994 he has been a professor of mathematics in the United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 .... He was a professor at Johns Hopkins University until 2002 when he became the first person to hold the Mina Rees Chair in mathematics at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer Lattice (group), lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. There is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
