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Belyi's Theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data. Quotients of the upper half-plane It follows that the Riemann surface in question can be taken to be the quotient :''H''/Γ (where ''H'' is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is ''not'' the conclusion that any such curve is a modular curve. Belyi functions A Belyi function is a holomorphic map from a compact Riemann surface ''S'' to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
