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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] |
Princeton, New Jersey
The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Princeton Township, New Jersey, Princeton Township, both of which are now defunct. As of the 2020 United States census, the borough's population was 30,681, an increase of 2,109 (+7.4%) from the 2010 United States census, 2010 census combined count of 28,572. In the 2000 United States census, 2000 census, the two communities had a total population of 30,230, with 14,203 residents in the borough and 16,027 in the township. Princeton was founded before the American Revolutionary War. The borough is the home of Princeton University, one of the world's most acclaimed research universities, which bears its name and moved to the community in 1756 from the educational institution's previous location in Newark, New Jersey, Newark. Although its associ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taniyama–Shimura–Weil Conjecture
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 le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shafarevich–Weil Theorem
In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by for local fields and by for global fields. Statement Suppose that ''F'' is a global field, ''K'' is a normal extension of ''F'', and ''L'' is an abelian extension of ''K''. Then the Galois group Gal(''L''/''F'') is an extension of the group Gal(''K''/''F'') by the abelian group Gal(''L''/''K''), and this extension corresponds to an element of the cohomology group H2(Gal(''K''/''F''), Gal(''L''/''K'')). On the other hand, class field theory gives a fundamental class in H2(Gal(''K''/''F''),''I''''K'') and a reciprocity law map from ''I''''K'' to Gal(''L''/''K''). The Shafarevich–Weil theorem states that the class of the extension Gal(''L''/''F'') is the image of the fundamental class under the homomorphism of cohomology groups induced by the reciprocity law map . Shafarevich stated his th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel–Weil Formula
In mathematics, the Siegel–Weil formula, introduced by as an extension of the results of , expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the order of the automorphism group of the lattice. For the constant terms this is essentially the Smith–Minkowski–Siegel mass formula In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism gr .... References * * * * Theorems in number theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oka–Weil Theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. Statement The Oka–Weil theorem states that if ''X'' is a Stein space and ''K'' is a compact \mathcal(X)-convex subset of ''X'', then every holomorphic function in an open neighborhood of ''K'' can be approximated uniformly on ''K'' by holomorphic functions on \mathcal(X) (in particular, by polynomials). Applications Since Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The Behnke–Stein theorem was originally proved using the Oka–Weil theorem. See also * Oka coherence theorem In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal_ of holomorphic functions on \mathbb^n (and subsequently the sheaf \mathcal_ of holomorphic functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mordell–Weil Theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elliptic curve E and K the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties. History The ''tangent-chord process'' (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(\mathbb)/2E(\mathbb) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(\mathbb) to be finitely generated; and it shows that the rank is fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mordell–Weil Group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K. It is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mordell–Weil grouppg 207. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of A(K) to the zero of the associated L-function at a special point. Examples Constructing explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve E/\mathbb. Let E be defined by the Weierstrass equationy^2 = x(x-6)(x+6)over the rational numbers. It has discriminant \Delta_E = 2^\cdot 3^6 (and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse–Weil Zeta Function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number ''p''. It is a global ''L''-function defined as an Euler product of local zeta functions. Hasse–Weil ''L''-functions form one of the two major classes of global ''L''-functions, alongside the ''L''-functions associated to automorphic representations. Conjecturally, these two types of global ''L''-functions are actually two descriptions of the same type of global ''L''-function; this would be a vast generalisation of the Taniyama-Weil conjecture, itself an important result in number theory. For an elliptic curve over a number field ''K'', the Hasse–Weil zeta function is conjecturally related to the group of rational points of the elliptic curve over ''K'' by the Birch and Swinnerton-Dyer conjecture. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse's Theorem On Elliptic Curves
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If ''N'' is the number of points on the elliptic curve ''E'' over a finite field with ''q'' elements, then Hasse's result states that :, N - (q+1), \le 2 \sqrt. The reason is that ''N'' differs from ''q'' + 1, the number of points of the projective line over the same field, by an 'error term' that is the sum of two complex numbers, each of absolute value \sqrt. This result had originally been conjectured by Emil Artin in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936. Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of ''E''. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve. Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil's Explicit Formula
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field. Riemann's explicit formula In his 1859 paper "On the Number of Primes Less Than a Given Magnitude" Riemann sketched an explicit formula (it was not fully proven until 1895 by von Mangoldt, see below) for the normalized prime-counting function which is related to the prime-counting function by :\pi_0(x) = \frac \lim_ \left ,\pi(x+h) + \pi(x-h)\,\right,, which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. His formula was given in terms of the related function :f(x) = \pi_0(x) + \frac\,\pi_0(x^) + \frac\,\pi_0(x^) + \cdots in which a prime power counts as of a prime. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
