Concept
Automorphic form
Related concepts (20)
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950 and an honorary Abel Prize in 2002. Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his three brothers, Sigmund and Henrik, were also mathematicians.
Congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.
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. A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient a0 in the Fourier series expansion (see q-expansion) This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter.
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics.
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. Gorō Shimura was born in Hamamatsu, Japan, on 23 February 1930. Shimura graduated with a B.A. in mathematics and a D.
Function of several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi.
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. For any complex number τ with Im(τ) > 0, let q = e2πiτ; then the eta function is defined by, Raising the eta equation to the 24th power and multiplying by (2π)12 gives where Δ is the modular discriminant.
André Weil
André Weil ('veɪ; ɑ̃dʁe vɛj; 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.