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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 a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister. The writer Sylvie Weil is his daughter. 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 Muslim University in India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he had taught himself Sanskrit in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Since the 17th century, Paris has been one of the world's major centres of finance, diplomacy, commerce, fashion, gastronomy, and science. For its leading role in the arts and sciences, as well as its very early system of street lighting, in the 19th century it became known as "the City of Light". Like London, prior to the Second World War, it was also sometimes called the capital of the world. The City of Paris is the centre of the Île-de-France region, or Paris Region, with an estimated population of 12,262,544 in 2019, or about 19% of the population of France, making the region France's primate city. The Paris Region had a GDP of €739 billion ($743 billion) in 2019, which is the highest in Europe. According to the Economist Intelli ... [...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) (i.e. 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 References Bibliography * * * * * * Further reading * – An example where Runge's theorem In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after t ... [...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 ... [...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. De ... [...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. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham–Weil Theorem
In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question. Let \mathcal F be a sheaf on a topological space X and \mathcal F^\bullet a resolution of \mathcal F by acyclic sheaves. Then : H^q(X,\mathcal F) \cong H^q(\mathcal F^\bullet(X)), where H^q(X,\mathcal F) denotes the q-th sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ... group of X with coefficients in \mathcal F. The De Rham–Weil theorem follows from the more general fact that derived functors may be computed using acyclic resolutions instead of simply injective resolutions. References * * * * {{DEFAULTSORT:De Rham-Weil theorem Homological algebra Sheaf theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
