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Cusp Form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient ''a''0 in the Fourier series expansion (see ''q''-expansion) :\sum a_n q^n. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation :z\mapsto z+1. For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as ''q'' → 0 is the limit in the upper half-plane as the imaginary part of ''z'' → ∞. Taking the quotient by the modular group, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying that a cus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goro Shimura
was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem. Biography Gorō Shimura was born in Hamamatsu, Japan, on 23 February 1930. Shimura graduated with a B.A. in mathematics and a D.Sc. in mathematics from the University of Tokyo in 1952 and 1958, respectively. After graduating, Shimura became a lecturer at the University of Tokyo, then worked abroad — including ten months in Paris and a seven-month stint at Princeton's Institute for Advanced Study — before returning to Tokyo, where he married Chikako Ishiguro. He then moved from Tokyo to join the faculty of Osaka University, but growing unhappy with his funding situation, he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuspidal Representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term ''cuspidal'' is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When the group is the general linear group \operatorname_2, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform ( newform) corresponds to a cuspidal representation. Formulation Let ''G'' be a reductive algebraic group over a number field ''K'' and let A denote the adeles of ''K''. The group ''G''(''K'') embeds diagonally in the group ''G''(A) by sending ''g'' in ''G''(''K'') to the tuple (''g''''p'')''p'' in ''G''(A) with ''g'' = ''g''''p'' for all (finite and infinite) primes ''p''. Let '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair. Here the group ''B'' is a Borel subgroup and ''N'' is the normalizer of a maximal torus contained in ''B''. The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraic groups. Parabolic subgroups Subgroups between a Borel subgroup ''B'' and the ambient group ''G'' are called parabolic subgroups. Parabolic subgroups ''P'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Eisenstein series for the modular group Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: :G_(\tau) = \sum_ \frac. This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Relation to modular invariants The modular invariants and of an elliptic curve are given by the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
