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List Of Things Named After André Weil
These are things named after André Weil (1906 – 1998), a French mathematician. *Bergman–Weil formula *Borel–Weil theorem *Chern–Weil homomorphism *Chern–Weil theory *De Rham–Weil theorem *Explicit formulae (L-function)#Weil's explicit formula, Weil's explicit formula *Hasse%27s_theorem_on_elliptic_curves#Hasse-Weil_Bound, Hasse-Weil bound *Hasse–Weil zeta function, and the related Hasse–Weil L-function *Mordell–Weil group *Mordell–Weil theorem *Oka–Weil theorem *Siegel–Weil formula *Shafarevich–Weil theorem *Taniyama–Shimura–Weil conjecture, now proved as the modularity theorem *Weil algebra *Weil–Brezin Map *Weil–Châtelet group *Weil cohomology *Weil conjecture (other) *Weil conjectures *Weil conjecture on Tamagawa numbers *Weil's criterion *Weil–Deligne group scheme *Weil distribution *Weil divisor *Weil group *Height_function#Weil_height, Weil height *Weil number *Weil pairing *Weil–Petersson metric *Weil reciprocity law *Weil repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the Bourbaki group, of which he was one of the principal founders. Life André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71. Simone Weil, who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil–Châtelet Group
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety ''A'' defined over a field ''K'' is the abelian group of principal homogeneous spaces for ''A'', defined over ''K''. named it for who introduced it for elliptic curves, and , who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent. It can be defined directly from Galois cohomology, as H^1(G_K,A), where G_K is the absolute Galois group of ''K''. It is of particular interest for local fields and global fields, such as algebraic number fields. For ''K'' a finite field, proved that the Weil–Châtelet group is trivial for elliptic curves, and proved that it is trivial for any connected algebraic group. See also The Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil–Petersson Metric
In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space ''T''''g'',''n'' of genus ''g'' Riemann surfaces with ''n'' marked points. It was introduced by using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson). Definition If a point of Teichmüller space is represented by a Riemann surface ''R'', then the cotangent space at that point can be identified with the space of quadratic differentials at ''R''. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric. Properties stated, and proved, that the Weil–Petersson metric is a Kähler metric. proved that it has negative holomorphic sectional, scalar, and R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing ''n'' of an elliptic curve ''E'', taking values in ''n''th roots of unity. More generally there is a similar Weil pairing between points of order ''n'' of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function. Formulation Choose an elliptic curve ''E'' defined over a field ''K'', and an integer ''n'' > 0 (we require ''n'' to be coprime to char(''K'') if char(''K'') > 0) such that ''K'' contains a primitive nth root of unity. Then the ''n''-torsion on E(\overline) is known to be a Cartesian product of two cyclic groups of order ''n''. The Weil pairing produces an ''n''-th root of unity :w(P,Q) \in \mu_n by mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Number
Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg *Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with the surname Weill, Weyl * Doctor Weil (''Mega Man Zero''), a fictional character from the video game series * Weil, Gotshal & Manges Weil, Gotshal & Manges LLP ( ) is an American law firm headquartered in New York City. Founded in 1931, it employs approximately 1,100 attorneys and reported annual revenues of over $1.8 billion, ranking it within''The American Lawyer'' AmLaw 10 ..., international law firm See also * * * Weill (other) {{disambiguation, geo ... [...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] |
Weil Group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite level" modifications of the Galois groups: if ''E''/''F'' is a finite extension, then the relative Weil group of ''E''/''F'' is ''W''''E''/''F'' = ''WF''/ (where the superscript ''c'' denotes the commutator subgroup). For more details about Weil groups see or or . Class formation The Weil group of a class formation with fundamental classes ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''W''''E''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''W''''E''/''F'' → Gal(''E''/''F'') → 1 corresponding (usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Distribution
Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg *Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with the surname Weill, Weyl * Doctor Weil (''Mega Man Zero''), a fictional character from the video game series * Weil, Gotshal & Manges Weil, Gotshal & Manges LLP ( ) is an American law firm headquartered in New York City. Founded in 1931, it employs approximately 1,100 attorneys and reported annual revenues of over $1.8 billion, ranking it within''The American Lawyer'' AmLaw 10 ..., international law firm See also * * * Weill (other) {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil–Deligne Group Scheme
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite level" modifications of the Galois groups: if ''E''/''F'' is a finite extension, then the relative Weil group of ''E''/''F'' is ''W''''E''/''F'' = ''WF''/ (where the superscript ''c'' denotes the commutator subgroup). For more details about Weil groups see or or . Class formation The Weil group of a class formation with fundamental classes ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''W''''E''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''W''''E''/''F'' → Gal(''E''/''F'') → 1 corresponding (usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil's Criterion
In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of prime number theory, as they apply to Dirichlet L-functions, and other more general global L-functions. A single statement thus combines statements on the complex zeroes of ''all'' Dirichlet L-functions. Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions ( Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Conjecture On Tamagawa Numbers
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number \tau(G) of a simply connected simple algebraic group defined over a number field is 1. In this case, ''simply connected'' means "not having a proper ''algebraic'' covering" in the algebraic group theory sense, which is not always the topologists' meaning. History calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture. Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
