automorphic number oor Nederlands
automorphic number
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(mathematics) A number whose decimal representation of its square ends in itself.
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number whose square ends in itself
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25 is a centered octagonal number, a centered square number, and an automorphic number.
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The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3.
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Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.
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New methods and techniques used in number theory: Functional equations arising from automorphic forms Analytic continuation (although not in the spirit of Weierstrass) Contour integration Fourier inversion.
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Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence.
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The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field.
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has an impact in number theory via automorphic forms and the Langlands program.[9]
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He is noted for many theorems including: Hölder's inequality, the Jordan–Hölder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group of real numbers, the classification of simple groups of order up to 200, the anomalous outer automorphisms of the symmetric group S6, and Hölder's theorem, which implies that the Gamma function satisfies no algebraic differential equation.
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Conjecturally there is just one essential type of global L-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of the Taniyama–Shimura conjecture, itself a very deep and recent result (as of 2009) in number theory.
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The theory of modular forms was developed in four periods: first in connection with the theory of elliptic functions, in the first part of the nineteenth century; then by Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable); then by Erich Hecke from about 1925; and then in the 1960s, as the needs of number theory and the formulation of the modularity theorem in particular made it clear that modular forms are deeply implicated.
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The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.
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It states that the nontrivial zeros of all automorphic L-functions lie on the critical line 1/2 + it with t a real number variable and i the imaginary unit.
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HistoryEdit The theory of modular forms was developed in four periods: first in connection with the theory of elliptic functions, in the first part of the nineteenth century; then by Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable); then by Erich Hecke from about 1925; and then in the 1960s, as the needs of number theory and the formulation of the modularity theorem in particular made it clear that modular forms are deeply implicated.
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