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Mathematical concept
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
Automorphic_L-function
Mathematical function on a space that is invariant under the action of some group
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
Automorphic_function
Type of Dirichlet series associated to number field extensions
incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far
Artin_L-function
Type of generalization of periodic functions in Euclidean space
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G {\displaystyle G} to the complex numbers
Automorphic_form
Conjectures connecting number theory and geometry
) Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional
Langlands_program
representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors
Rankin–Selberg_method
Meromorphic function on the complex plane
function, π {\displaystyle \textstyle \pi } denotes the automorphic number, and d {\displaystyle \textstyle d} denotes the degree of the L-function mentioned
L-function
Mathematical concept
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, standard refers
Standard_L-function
Topics referred to by the same term
mathematics Automorphic form, in mathematics Automorphic representation, in mathematics Automorphic L-function, in mathematics Automorphism, in mathematics
Automorphic
Mathematical concept
equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology
Shimura_variety
Axiomatic definition of a class of L-functions
class is equal to class of automorphic L-functions. Primitive functions are expected to be associated with irreducible automorphic representations. It is
Selberg_class
as automorphic L-functions, and hence should be part of the Selberg class. There are also conjectures concerning the values of these L-functions at integers
Motivic_L-function
Mathematical conjectures in class field theory
Borel, A. (1979). "Automorphic L-functions". In Borel, A.; Casselman, W. (eds.). Automorphic Forms, Representations, and L-functions. Proceedings of Symposia
Local_Langlands_conjectures
Canadian mathematician
the Hasse–Weil zeta functions of certain Shimura varieties are among the L {\displaystyle L} -functions arising from automorphic forms. The functoriality
Robert_Langlands
Exploring properties of the integers with complex analysis
describing the density of the zeros on the critical line. Automorphic L-function Automorphic form Langlands program Maier's matrix method Apostol 1976
Analytic_number_theory
mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a
Langlands–Shahidi_method
American mathematician and professor (born 1973)
is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory. Soundararajan grew up in Chennai
Kannan_Soundararajan
Theorem in number theory
transforms of weight 2 modular forms or a product of analogous automorphic L-functions. Eichler, Martin (1954), "Quaternäre quadratische Formen und die
Eichler–Shimura congruence relation
Eichler–Shimura_congruence_relation
Unsolved problem in mathematics
more generally, automorphic forms. The name of the conjecture comes from Srinivasa Ramanujan, who proposed it for Ramanujan tau function, and Hans Petersson
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
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
Maass_wave_form
Group controlling representation theory
theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by Langlands (1967) in a letter to A. Weil. The L-group is
Langlands_dual_group
In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms
Automorphic_factor
Conjecture on zeros of the zeta function
all Automorphic L-functions, such as Mellin transforms of Hecke eigenforms. The Riemann hypothesis for Selberg class extends it rather for functions satisfying
Riemann_hypothesis
Riemann hypothesis. It states that the non-trivial zeros of all automorphic L-functions lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with
Grand_Riemann_hypothesis
MR 0088511 Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982. Sunada, T., L-functions in geometry and some applications,
Selberg_zeta_function
On the reciprocity law in algebraic number fields
reciprocity involving Artin L-functions and automorphic L-functions: for finite number field extension L / K {\displaystyle L/K} , let ρ {\displaystyle
Hilbert's_ninth_problem
Iranian mathematician
Mathematics at Purdue University in the U.S. He is known for a method of automorphic L-functions which is now known as the Langlands–Shahidi method. Shahidi graduated
Freydoon_Shahidi
Mathematical functions that quantify complexity
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of
Height_function
Japanese mathematician (1930–2019)
equivalence between motivic and automorphic L-functions postulated in the Langlands program could be tested: automorphic forms realized in the cohomology
Goro_Shimura
Selberg class of Dirichlet series equal to class of automorphic L-functions? Hardy–Littlewood zeta function conjectures Keating–Snaith conjecture concerning
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Analytic function on the upper half-plane with a certain behavior under the modular group
growth condition. A modular form is a special case of an automorphic form, which are functions defined on Lie groups that transform nicely with respect
Modular_form
Israeli mathematician (1929–2009)
geometry. His main contribution and impact was in the area of automorphic forms and L-functions. For the last 30 years of his life he suffered from Parkinson's
Ilya_Piatetski-Shapiro
American mathematician
series on the metaplectic group and non vanishing theorems for automorphic L-functions and their derivatives, Annals of Mathematics, vol. 131, 1990, pp
Jeffrey_Hoffstein
Mathematical function associated to algebraic varieties
L-functions, alongside the L-functions associated to automorphic representations. Conjecturally, these two types of global L-functions are actually two descriptions
Hasse–Weil_zeta_function
Statement in number theory
JSTOR 2153215. Lagarias, Jeffrey C. (2004). "Li coefficients for automorphic L-functions". Annales de l'Institut Fourier. 57 (2007): 1689–1740. arXiv:math
Li's_criterion
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional
Functional equation (L-function)
Functional_equation_(L-function)
the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of
Cuspidal_representation
Completes the Langlands program for general linear groups over algebraic function fields
program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of
Lafforgue's_theorem
American-Israeli mathematician
Explicit constructions of automorphic L-functions. 1987. with Freydoon Shahidi: Analytic properties of automorphic L-functions. Academic Press. 1988. later
Stephen_Gelbart
Indian mathematician (born 1976)
is known for his contributions to the sub-convexity problem for automorphic L-functions. In a series of papers published in 2015 he introduced a new approach
Ritabrata_Munshi
Modular form
Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994. ISBN 0-691-08092-5 Gelbart, Stephen, Automorphic Forms on Adele Groups
Cusp_form
36 mathematical problems stated in 1955
this form is an elliptic differential of the field of associated automorphic functions. Now, going through these observations backward, is it possible
Taniyama's_problems
American mathematician
for automorphic L-functions, Inventiones Mathematicae, 112, 1–8. Duke, W., Friedlander, J., Iwaniec, H. (1994) Bounds for automorphic L-functions II,
William_Duke_(mathematician)
Mathematical formula in harmonic analysis
been a standard tool for studying analytic properties of automorphic forms and their L-functions. There have been numerous results coming out the Voronoi
Voronoi_formula
American mathematician
works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous dynamics. Kontorovich
Alex_Kontorovich
Special functions of several complex variables
the theta series to automorphic forms with respect to arbitrary Fuchsian groups. In the following, three important theta function values are to be derived
Theta_function
Mathematic theory
Zeta functions of simple algebras, Lect. Notes Math., vol. 260, Springer Goldfeld, Dorian; Hundley, Joseph (2011), Automorphic representations of L-functions
Tate's_thesis
Conjecture in the representation theory of Lie groups
nonvanishing of the central value of the Rankin-Selberg L-functions II", Automorphic Representations, L-Functions and Applications: Progress and Prospects, Berlin:
Gan–Gross–Prasad_conjecture
In projective geometry, a bijection between projective spaces that preserves collinearity
are projective linear transformations (also known as homographies) and automorphic collineations. For projective spaces coming from a linear space, the
Collineation
Function defined by a hypergeometric series
is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r). The monodromy
Hypergeometric_function
Korean educator (born 1978)
(2016). "Sato–Tate theorem for families and low-lying zeros of automorphic L-functions". Inventiones Mathematicae. 203 (1): 1–177. Bibcode:2016InMat.203
Sug_Woo_Shin
Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7. Bump, Daniel (1998). Automorphic Forms and Representations
Schwartz–Bruhat_function
Mathematical theorem
theory of automorphic forms and in analytic number theory. The trace formula is also central to the analytic theory of the Selberg zeta function. It can
Selberg_trace_formula
expression appearing in the functional equation of an Artin L-function. Suppose that L {\displaystyle L} is a finite Galois extension of the local field K {\displaystyle
Artin_conductor
American mathematician
representation theory, and automorphic forms. His early research focused on integral representations of automorphic L-functions. In joint work with Hervé
Solomon_Friedberg
all of the Pontryagin duality type, rather than needing more general automorphic representations. That reflects a good understanding of their Tate modules
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Polish-American mathematician (born 1947)
Jonathan D. (1998). "Book Review: Automorphic forms on S L 2 ( R ) {\displaystyle SL_{2}(\mathbf {R} )} by A. Borel, Automorphic forms and representations by
Henryk_Iwaniec
Conformal mappings in complex analysis
Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function. Let πα, πβ
Schwarz_triangle_function
Russian mathematician
doctorate (higher doctoral degree) with dissertation Spectral theory of automorphic functions (Russian). He was a visiting scholar at IHES, at the University
Alexei_Venkov
Class of complex vector function
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg
Jacobi_form
Chinese mathematician (born 1981)
Peking University working in number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory
Xinyi_Yuan
Mathematical theorem
L-functions, converse theorems, and functoriality for GLn", in Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (eds.), Lectures on automorphic L-functions
Multiplicity-one_theorem
Concept in number theory
algebraic, analytic, and growth conditions. In this setting, automorphic L {\displaystyle L} -functions can often be described by integrals over adelic groups
Adele_ring
Mathematical function
Mathematika. 1: 4. doi:10.1112/S0025579300000462. Bump, Daniel (1998), Automorphic Forms and Representations, Cambridge University Press, ISBN 0-521-55098-X
Dedekind_eta_function
Concept in mathematics
Shtuka", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis
Drinfeld_module
Mathematician at the University of Minnesota
value of the Rankin-Selberg L-functions. J. Amer. Math. Soc. 17 (2004), no. 3, 679–722. On the fundamental automorphic L-functions of SO(2n+1). Int. Math.
Dihua_Jiang
Mathematician
Galois representations and some automorphic forms. The "naturalness" is guaranteed by the essential coincidence of L-functions. However, this condition is
Vladimir_Drinfeld
Special function of two variables
1962. Zagier, D. (1981). "Eisenstein series and the Riemann zeta-function". Automorphic Forms, Representation Theory and Arithmetic. Springer Berlin, Heidelberg
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
American mathematician
number theory, arithmetic geometry, and automorphic forms, in particular, Hilbert modular forms and zeta functions of Shimura varieties. He was a visiting
Don_Blasius
working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his
Hervé_Jacquet
Iterative algorithm on numbers
sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle K_{b}(n)=\alpha -\beta } is the Kaprekar
Kaprekar's_routine
American mathematician (born 1953)
he gave the 2009 Erwin Schrödinger Lecture). Cogdell works on L-functions, automorphic forms (within the context of the Langlands program), and analytic
James_Cogdell
American mathematician (born 1947)
Mathematical Society. Goldfeld, Dorian; Hundley, Joseph (2011). Automorphic Representations and L-Functions for the General Linear Group, Volume 1. Cambridge University
Dorian_M._Goldfeld
Type of vector space
Prasanna, Kartik; Venkatesh, Akshay (2021). "Automorphic cohomology, motivic cohomology, and the adjoint L-function". Astérisque. 428. ISBN 978-2-85629-943-2
Hecke_algebra
Semitopological group in abstract algebra
"Automorphic forms and automorphic representations". In Borel, Armand; Casselman, William (eds.). Automorphic Forms, Representations and L-functions.
Adelic_algebraic_group
Iranian mathematician
Takloo-Bighash computed the local factors of spinor L-function attached to generic automorphic forms on the symplectic group GSp(4).[citation needed]
Ramin_Takloo-Bighash
Chinese mathematician (born 1981)
Academy of Arts and Sciences in 2023. "Automorphic period and the central value of Rankin-Selberg L-function", J. Amer. Math. Soc. 27 (2014), 541–612
Wei_Zhang_(mathematician)
Mathematical theory
G. (1983). "Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2)". American Journal
Geometric Langlands correspondence
Geometric_Langlands_correspondence
French mathematician
Langlands program in the function field case", namely for establishing the Langlands Correspondence (the direction from automorphic forms to Galois representations)
Vincent_Lafforgue
Type of theorem in automorphic forms
representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved. The first converse
Converse_theorem
Theorem in abstract algebra
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Mathematical conjecture
extended to correlations of more than two zeros, and also to zeta functions of automorphic representations (Rudnick & Sarnak 1996). In 1982 a student of Montgomery's
Montgomery's pair correlation conjecture
Montgomery's_pair_correlation_conjecture
Branch of mathematics that studies abstract algebraic structures
ISBN 978-0-8218-0288-5. Borel, Armand; Casselman, W. (1979), Automorphic Forms, Representations, and L-functions, American Mathematical Society, ISBN 978-0-8218-1435-2
Representation_theory
American mathematician (1942–2012)
mathematician who worked on group representations, automorphic forms, the Siegel–Weil formula, and Langlands L-functions. Rallis received a B.A. in 1964 from Harvard
Stephen_Rallis
Ramanujan–Petersson conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selberg trace formula Artin conjecture Sato–Tate conjecture Langlands
List_of_number_theory_topics
Mathematician
Freydoon; Soudry, David (eds.). Advances in the Theory of Automorphic Forms and Their L-functions. Contemporary Mathematics. Vol. 664. p. i. David Soudry
David_Soudry
American mathematician (1886–1967)
reputation. In 1915 Ford published An Introduction to the Theory of Automorphic Functions as Edinburgh Mathematical Tract # 6. Returning to Harvard in 1917
Lester_R._Ford
Mathematical object
-dimensional complex representations of L F {\displaystyle L_{F}} and, in the global case, the cuspidal automorphic representations of GL n ( A F ) {\displaystyle
Langlands_group
Number, approximately 3.14
the Jacobi theta function an automorphic form, meaning that it transforms in a specific way. Certain identities hold for all automorphic forms. An example
Pi
Mathematical constant
Fibonacci zeta function", in Prasad, D.; Rajan, C. S.; Sankaranarayanan, A.; Sengupta, J. (eds.), Automorphic representations and L-functions, Tata Institute
Reciprocal_Fibonacci_constant
Arithmetic operation
function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit. More precisely, consider the function f
Exponentiation
(functions related to) zeta functions associated to an arithmetic scheme have been suggested to correspond to automorphicity of the related L-function
Mean-periodic_function
Artin's L function is entire (holomorphic on the entire complex plane). automorphic form An automorphic form is a certain holomorphic function. Bézout's
Glossary_of_number_theory
Numbers obtained by adding the two previous ones
F\geq 1.} Instead using the floor function gives the largest index of a Fibonacci number that is not greater than F: n l a r g e s t ( F ) = ⌊ log φ 5
Fibonacci_sequence
Particular kind of exponential sum
the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics
Kloosterman_sum
American mathematician
Jonathan D. (1998). "Book Review: Automorphic forms on S L 2 ( R ) {\displaystyle SL_{2}(\mathbf {R} )} by A. Borel, Automorphic forms and representations by
Daniel_Bump
Mathematical theorem
idèle class group of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles
Artin_reciprocity
New Zealand mathematician
in Mathematics, 1986 Editor with Laurent Clozel, Automorphic Forms, Shimura Varieties and L-Functions, 2 volumes, Elsevier 1988 (Conference University
James_Milne_(mathematician)
View of mathematicians to consolidate two or more theories into a more generalized one
theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands. His principle of functoriality with respect to the L-group has
Unifying theories in mathematics
Unifying_theories_in_mathematics
Mathematical structure
applies to elliptic curves and elliptic functions; and in fact to curves of genus > 1 and automorphic functions.) The properties of algebraic curves, such
Transcendental_curve
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
Boy/Male
Indian
Lord of majesty and generosity
Girl/Female
Indian
Pl of hazz, Fortune, Good l
Male
Swedish
Swedish form of Greek Paulos, PÃ…L means "small."
Male
Norwegian
Norwegian variant form of Scandinavian Njal, NJÃ…L means "champion."
Girl/Female
Assamese, British, Gujarati, Hindu, Indian, Kannada, Malay, Malayalam, Marathi, Mythological, Oriya, Sindhi, Tamil
Like a Goddess; Daughter of Shukraacharya; L
Male
Dutch
, God's judge.
Boy/Male
Indian, Sanskrit
Miner; L Digger
Girl/Female
African, Arabic, Australian, Danish, German, Muslim, Pashtun, Swahili
Pure; L; Holy; Clean; Dean
Male
Irish
Irish Gaelic form of Greek MichaÄ“l, MÃCHEÃL means "who is like God?"
Male
French
French name derived from Latin natalis dies, NOËL means "day of birth."
Male
Hungarian
Hungarian form of Greek Paulos, PÃL means "small."
Male
Irish
Irish form of Greek Paulos, PÓL means "small."
Male
French
French form of Hebrew Rephael, RAPHAËL means "healed of God" or "whom God has healed."
Male
French
Masculine form of French Gaëlle, GAËL means "holy and generous."
Male
Hungarian
Hungarian form of Roman Latin Cornelius, KORNÉL means "of a horn."
Boy/Male
Muslim
Lord of majesty and generosity
Male
French
French form of Greek Ioel (Hebrew Yowel), JOËL means "Jehovah is God" or "to whom Jehovah is God."
Male
Scottish
Scottish form of Latin Paulus, PÀL means "small."
Girl/Female
Muslim
Pl of hazz, Fortune, Good l
Boy/Male
Irish
Rooster.
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lakshman
Surname or Lastname
English
English : occupational name for a maker of pouches, from the plural of Middle English crippes ‘pouch’.English : metathesized form of Crisp.German : variant spelling of Krips, a variant of Krebs.
Boy/Male
Hindu, Indian, Sanskrit
In the Heart
Girl/Female
Arabic, Muslim
Red; Beautiful
Surname or Lastname
English
English : patronymic from a variant of the personal name Gibbon, a pet form of Gibb.
Boy/Male
Indian
Lord Krishna
Girl/Female
Muslim English
Created. Produced.
Surname or Lastname
English (mainly Gloucestershire), Dutch, and German (also Türk)
English (mainly Gloucestershire), Dutch, and German (also Türk) : from Middle English, Old French turc, Middle High and Low German Turc ‘Turk’, from Turkish türk. In theory this could be an ethnic name but, both in England and northwest Europe, it is generally a nickname for a person with black hair and a swarthy complexion or a cruel, rowdy, or unruly person. The Dutch and German surname also represents a house name, derived from the use of a picture of a Turk as a house sign. It is also found as a nickname for someone who had taken part in the wars against the Turks.English : from a medieval personal name, a back-formation from Turkel, misanalyzed as containing the Old French diminutive suffix -el.Scottish : reduced Anglicized form of Gaelic Mac Tuirc, a patronymic from the byname Torc ‘boar’.Jewish (Ashkenazic) : ethnic name denoting someone from Turkey or anywhere in the Ottoman Empire, or a nickname for someone thought to resemble a Turk.Americanized form of the Greek ethnic name Tourkos ‘Turk’. See also Turco.
Girl/Female
Indian
Good tiding
Boy/Male
Muslim
Diminutive of Hasan, Beautiful
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
n.
Automorphic characterization.
n.
A symbol representing fifty units, as 50, or l.
L. catechunenus, Gr.
One who is receiving rudimentary instruction in the doctrines of Christianity; a neophyte; in the primitive church, one officially recognized as a Christian, and admitted to instruction preliminary to admission to full membership in the church.
n.
Any small leguminous plant of the genus Lathyrus, especially L. Nissolia.
a.
Relating to Casserio (L. Gasserius), the discover of the Gasserian ganglion.
n.
The name of the Greek letter /, /, corresponding with the English letter L, l.
n.
A weed of the genus Lamium (L. amplexicaule) with deeply crenate leaves.
n.
A large stork of the genus Leptoptilos (formerly Ciconia), esp. the African species (L. crumenifer), which furnishes plumes worn as ornaments. The Asiatic species (L. dubius, or L. argala) is the adjutant. See Adjutant.
a.
Patterned after one's self.
a.
Of or pertaining to allomorphism.
n.
See L.
v. t.
To betray; to show. [L.]
n.
An extension at right angles to the length of a main building, giving to the ground plan a form resembling the letter L; sometimes less properly applied to a narrower, or lower, extension in the direction of the length of the main building; a wing.
n.
A short right-angled pipe fitting, used in connecting two pipes at right angles.