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- Automorphic forms

Jacquet–Langlands correspondence

In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL2 and its twisted forms, proved by Jacquet and Langlands in their book Automorphic Forms on GL(2

Miyawaki lift

The Miyawaki lift or Ikeda–Miyawaki lift or Miyawaki–Ikeda lift, is a mathematical lift that takes two Siegel modular forms to another Siegel modular form. Miyawaki conjectured the existence of this l

Kuznetsov trace formula

In analytic number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula. The Kuznetsov or relative trace formula connects Kloosterman sums at a deep level with the spectr

Θ10

In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field. introduced θ10 for

Fundamental lemma (Langlands program)

In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was co

Harmonic Maass form

In mathematics, a weak Maass form is a smooth function on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hype

Shimura variety

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebra

Koecher–Maass series

In mathematics, a Koecher–Maass series is a type of Dirichlet series that can be expressed as a Mellin transform of a Siegel modular form, generalizing Hecke's method of associating a Dirichlet series

Klingen Eisenstein series

In mathematics, a Klingen Eisenstein series is a Siegel modular form of weight k and degree g depending on another Siegel cusp form f of weight k and degree r

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 o

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifol

Schottky form

In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky as a degree 16 polynomial in the Thetanullwerte of genus 4.

Theta constant

In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to

Siegel parabolic subgroup

In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group who

Siegel theta series

In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice.

Drinfeld upper half plane

In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld. It is defined to be P1(C)\P1(F∞), where F i

Selberg's 1/4 conjecture

In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by Selberg , states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subg

Teichmüller modular form

In mathematics, a Teichmüller modular form is an analogue of a Siegel modular form on Teichmüller space.

Whittaker model

In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space

Maass wave form

In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar wa

Picard modular surface

In mathematics, a Picard modular surface, studied by Picard, is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group.Picard modular surfaces are some of the sim

Kleinian group

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of th

Picard modular group

In mathematics, a Picard modular group, studied by Picard, is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and J is a hermi

Siegel Eisenstein series

In mathematics, a Siegel Eisenstein series (sometimes just called an Eisenstein series or a Siegel series) is a generalization of Eisenstein series to Siegel modular forms. gave an explicit formula fo

Lax pair

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs

Arthur's conjectures

In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made

Multiplicity-one theorem

In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question i

Hilbert's twenty-second problem

Hilbert's twenty-second problem is the penultimate entry in the celebrated list of 23 Hilbert problems compiled in 1900 by David Hilbert. It entails the uniformization of analytic relations by means o

Poincaré series (modular form)

In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several com

Voronoi formula

In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by on either side. It can be regarded as a Poisson summation formula

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 Rober

Local trace formula

In mathematics, the local trace formula is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation of G(F) on the discrete part of L2(G(F)), for G a red

Siegel upper half-space

In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is p

Hilbert modular form

In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes satisf

Arthur–Selberg trace formula

In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a lo

Gan–Gross–Prasad conjecture

In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The probl

Langlands–Shahidi method

In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Sel

Converse theorem

In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem s

Selberg trace formula

In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where

Langlands dual group

In representation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G (also called the L-group of G) is a group that controls the representation theory of G. If G i

Local Langlands conjectures

In mathematics, the local Langlands conjectures, introduced by Robert Langlands , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive a

Endoscopic group

In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands in his work on the stable trace formula. Roughly speaking, an endoscopic group H of G is a quasi-spl

Igusa group

In mathematics, an Igusa group or Igusa subgroup is a subgroup of the Siegel modular group defined by some congruence conditions. They were introduced by Igusa.

Automorphic L-function

In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensio

Schwarz triangle function

In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs fo

Lafforgue's theorem

In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic

Siegel operator

In mathematics, the Siegel operator is a linear map from (level 1) Siegel modular forms of degree d to Siegel modular forms of degree d − 1, generalizing taking the constant term of a modular form. Th

Kirillov model

In mathematics, the Kirillov model, studied by Kirillov, is a realization of a representation of GL2 over a local field on a space of functions on the local field. If G is the algebraic group GL2 and

Siegel modular form

In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds cons

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