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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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mathematical conjectures in class field theory
Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure
Local_Langlands_conjectures
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7. Bump, Daniel (1998). Automorphic Forms and Representations
Schwartz–Bruhat_function
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)
Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994. ISBN 0-691-08092-5 Gelbart, Stephen, Automorphic Forms on Adele Groups
Cusp_form
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
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
works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous dynamics. Kontorovich
Alex_Kontorovich
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)
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
the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of
Cuspidal_representation
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mathematician
Galois representations and some automorphic forms. The "naturalness" is guaranteed by the essential coincidence of L-functions. However, this condition is
Vladimir_Drinfeld
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
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
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)
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
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
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
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
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
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)
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
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
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
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
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
Ramanujan–Petersson conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selberg trace formula Artin conjecture Sato–Tate conjecture Langlands
List_of_number_theory_topics
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
Number, approximately 3.14
Jacobi theta function an automorphic form, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example
Pi
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
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
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
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
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
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
American mathematician
his thesis was the special values of L-functions. Stevens’ research specialties are number theory, automorphic forms, and arithmetic geometry. He has
Glenn_H._Stevens
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
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
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
Product of an integer with itself
3066501376, both ending in 376. (The numbers 5, 6, 25, 76, etc. are called automorphic numbers. They are sequence A003226 in the OEIS.) In base 10, the last
Square_number
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
Relates rational elliptic curves to modular forms
Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form)
Modularity_theorem
Japanese mathematician (1930–2020)
contributions include works on p-adic L functions and real-analytic automorphic forms. His work on p-adic L-functions, later recognised as an aspect of Iwasawa
Tomio_Kubota
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
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
American mathematician (1950–2025)
particularly the Gross–Zagier theorem on L-functions of elliptic curves, and related topics in algebraic geometry, automorphic forms, and representation theory
Benedict_Gross
Number used for counting
a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves
Natural_number
Series representing modular forms
modular group, Eisenstein series can be generalized in the theory of automorphic forms. Let τ {\displaystyle \tau } be a complex number with strictly
Eisenstein_series
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
Mathematical conjecture about elliptic curves
not integral. Taylor, Richard (2008). "Automorphy for some l-adic lifts of automorphic mod l Galois representations. II". Publ. Math. Inst. Hautes Études
Sato–Tate_conjecture
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
Male
French
French form of Hebrew Rephael, RAPHAËL means "healed of God" or "whom God has healed."
Boy/Male
Indian
Lord of majesty and generosity
Male
Irish
Irish form of Greek Paulos, PÓL means "small."
Boy/Male
Irish
Rooster.
Girl/Female
Muslim
Pl of hazz, Fortune, Good l
Girl/Female
Assamese, British, Gujarati, Hindu, Indian, Kannada, Malay, Malayalam, Marathi, Mythological, Oriya, Sindhi, Tamil
Like a Goddess; Daughter of Shukraacharya; L
Male
Hungarian
Hungarian form of Roman Latin Cornelius, KORNÉL means "of a horn."
Girl/Female
Indian
Pl of hazz, Fortune, Good l
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
Swedish
Swedish form of Greek Paulos, PÃ…L means "small."
Boy/Male
Indian, Sanskrit
Miner; L Digger
Male
Dutch
, God's judge.
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
Norwegian
Norwegian variant form of Scandinavian Njal, NJÃ…L means "champion."
Male
Scottish
Scottish form of Latin Paulus, PÀL means "small."
Girl/Female
African, Arabic, Australian, Danish, German, Muslim, Pashtun, Swahili
Pure; L; Holy; Clean; Dean
Male
French
Masculine form of French Gaëlle, GAËL means "holy and generous."
Male
Irish
Irish Gaelic form of Greek MichaÄ“l, MÃCHEÃL means "who is like God?"
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
Female
Hebrew
Variant spelling of Hebrew Aminta, AMINTAH means "defender."
Boy/Male
Indian, Marathi
Courage and Patience
Boy/Male
British, English
From the Cottage on the Winding Path
Boy/Male
Hindu, Indian
Victorious
Girl/Female
Greek
Daffodil.
Boy/Male
Hindu
Name of Lord Shiva
Girl/Female
Muslim/Islamic
Prophets (PBUH) daughter
Girl/Female
English
meaning "From St. Denis.".
Boy/Male
Afghan, Arabic, French, Hebrew, Hindu, Indian, Muslim, Tamil
Comforter; Form of Raphael; God has Healed; Another Name for God; Lofty; Exalted; Delicate
Boy/Male
Hebrew
Witness of God.
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
AUTOMORPHIC L-FUNCTION
n.
A symbol representing fifty units, as 50, or l.
n.
The name of the Greek letter /, /, corresponding with the English letter L, l.
n.
A short right-angled pipe fitting, used in connecting two pipes at right angles.
n.
Any small leguminous plant of the genus Lathyrus, especially L. Nissolia.
n.
A weed of the genus Lamium (L. amplexicaule) with deeply crenate leaves.
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.
See L.
v. t.
To betray; to show. [L.]
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.
Relating to Casserio (L. Gasserius), the discover of the Gasserian ganglion.
a.
Of or pertaining to allomorphism.
n.
Automorphic characterization.
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.
a.
Patterned after one's self.