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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 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
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
Conjectures connecting number theory and geometry
of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by the Canadian mathematician Robert
Langlands_program
Number whose square ends in the same digits
{\displaystyle k} digits is an automorphic number if n {\displaystyle n} is a fixed point of the polynomial function f ( x ) = x 2 {\displaystyle f(x)=x^{2}}
Automorphic_number
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
British mathematician and historian of science (1873–1956)
a century. Throughout his career, he wrote papers on automorphic functions and special functions in pure mathematics as well as on electromagnetism, general
E._T._Whittaker
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
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
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
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
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
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
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
Concept in information theory
chain model, the Rényi entropy as a function of α can be calculated explicitly because it is an automorphic function with respect to a particular subgroup
Rényi_entropy
Kind of complex manifold
condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map
Complex_torus
Type of Dirichlet series associated to number field extensions
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
classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms
Parabolic_induction
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
American mathematician
York City) is an American mathematician, specializing in number theory, automorphic forms, and cryptography. Hoffstein graduated with a bachelor's degree
Jeffrey_Hoffstein
Distance from zero to a number
Press. ISBN 0-12-622760-8. Siegel, Carl Ludwig (1942). "Note on automorphic functions of several variables". Annals of Mathematics. Second Series. 43
Absolute_value
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
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
Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7. Bump, Daniel (1998). Automorphic Forms and Representations
Schwartz–Bruhat_function
German mathematician (1911–1992)
was also concerned with automorphic functions in several variables, Siegel modular functions, and associated zeta functions. Maass, Hans (1949), "Über
Hans_Maass
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
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
Arithmetical function
Weisstein, Eric W. "Dedekind Function". MathWorld. Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25,
Dedekind_psi_function
German mathematician (1849–1925)
especially equations he invented, satisfied by elliptic modular functions and automorphic functions. Klein showed that the modular group moves the fundamental
Felix_Klein
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
Algebraic variety
Shimura, Goro (1994) [1971], Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton
Modular_curve
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
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
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 Theory
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
23 mathematical problems stated in 1900
monodromy group Uniformization of analytic relations by means of automorphic functions Further development of the methods of the calculus of variations
Hilbert's_problems
American mathematician
worked on automorphic functions and introduced Atkin–Lehner theory. Lehner, Joseph (1964), Discontinuous groups and automorphic functions, Mathematical
Joseph_Lehner
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
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
Conjecture on zeros of the zeta function
2009-03-27 Zagier, Don (1981), "Eisenstein series and the Riemann zeta function", Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata
Riemann_hypothesis
English mathematician
automorphic functions, and functional equations. In 1894 Dixon wrote The Elementary Properties of the Elliptic Functions. Certain elliptic functions (meromorphic
Alfred_Cardew_Dixon
German mathematician (1826–1866)
and poles) of a Riemann surface. According to Detlef Laugwitz, automorphic functions appeared for the first time in an essay about the Laplace equation
Bernhard_Riemann
Formula in analytic number theory
was found by Kuznetsov while studying the growth of weight zero automorphic functions. Using estimates on Kloosterman sums he was able to derive estimates
Kuznetsov_trace_formula
On uniformization of analytic relations
It entails the uniformization of analytic relations by means of automorphic functions. The entirety of the original problem statement is as follows: As
Hilbert's twenty-second problem
Hilbert's_twenty-second_problem
German mathematician
partition function. He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to number theory, automorphic forms
Jan_Hendrik_Bruinier
Theorem in complex analysis
doi:10.1016/j.jmaa.2017.02.071. Weiss, M. "Pólya's Shire Theorem for Automorphic Functions". Geometriae Dedicata 100, 85–92 (2003). https://doi.org/10.1023/A:1025855513977
Pólya's_shire_theorem
American mathematician
Solomon Friedberg (born 1958) is an American mathematician specializing in automorphic forms, representation theory, and number theory. Friedberg received his
Solomon_Friedberg
Interference phenomenon of waves
Merzon, Anatoli (eds.), Stationary Diffraction by Wedges: Method of Automorphic Functions on Complex Characteristics, Cham: Springer International Publishing
Diffraction
Canadian mathematician
web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received
Robert_Langlands
Geometric concept of a 2D space with "points at infinity" adjoined
theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity
Projective_plane
Generalization of the 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
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
Chinese mathematician
Shimura, Gorō (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, N.J: Princeton University Press. ISBN 978-0-691-08092-5
Yunqing_Tang
Axiomatic definition of a class of L-functions
equal to the class of automorphic L-functions. Primitive functions are expected to be associated with irreducible automorphic representations. It is
Selberg_class
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
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
Mathematic theory
ISBN 978-3-540-05797-0. Goldfeld, Dorian; Hundley, Joseph (2011). Automorphic Representations of L-Functions for the General Linear Group. Cambridge Studies in Advanced
Tate's_thesis
American mathematician
referred book on scattering theory titled Scattering Theory for Automorphic Functions. Phillips received the 1997 Leroy P. Steele Prize for Lifetime Achievement
Ralph_S._Phillips
German mathematician (1861–1930)
his work in complex analysis, especially on elliptic, modular and automorphic functions. He was one of the main collaborators of Felix Klein, with whom
Robert_Fricke
ISSN 0012-7094, MR 0534060 Lehner, Joseph (1964), Discontinuous Groups and Automorphic Functions, Mathematical Surveys and Monographs, vol. 8, American Mathematical
Schottky_group
German mathematician (1938–2025)
ISSN 0012-9593. (online). Harder, G. (1974). "Chevalley Groups Over Function Fields and Automorphic Forms". The Annals of Mathematics. 100 (2). JSTOR: 249–306
Günter_Harder
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
German mathematician (1862–1943)
monodromy group. 22. Uniformization of analytic relations by means of automorphic functions. 23. Further development of the methods of the calculus of variations
David_Hilbert
Complex multiplication field
Shimura, Goro (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan. Vol. 11. Princeton
CM-field
Arithmetic operation
exponent. For example, if f is a function whose valued can be multiplied, f∘n denotes exponentiation with respect to function composition, whereas fn may denote
Exponentiation
Matrices satisfying a differential equation
1002/cpa.3160210503, OSTI 4522657 archive P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions[1], (1976) Princeton University Press.
Lax_pair
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
acting on a domain D and H(z) is any meromorphic function on D, then one obtains an automorphic function by averaging over Γ: ∑ γ ∈ Γ H ( γ ( z ) ) . {\displaystyle
Poincaré series (modular form)
Poincaré_series_(modular_form)
Kronecker delta function, S is the Kloosterman sum and J is the Bessel function of the first kind. Henryk Iwaniec: Topics in Classical Automorphic Forms. Graduate
Petersson_trace_formula
On the action of ideles of imaginary quadratic fields on the values of modular functions
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Tokyo:
Shimura's_reciprocity_law
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
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
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
Special function of two variables
1962. Zagier, D. (1981). "Eisenstein series and the Riemann zeta-function". Automorphic Forms, Representation Theory and Arithmetic. Berlin, Heidelberg:
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Matrix equal to its transpose
JSTOR 2371774, Lemma 1, page 12 Hua, L.-K. (1944), "On the theory of automorphic functions of a matrix variable I–geometric basis", Amer. J. Math., 66 (3):
Symmetric_matrix
Whittaker, E. T. (January 1898). "On the connexion of algebraic functions with automorphic functions". Proceedings of the Royal Society of London. 63 (389–400):
Bibliography of E. T. Whittaker
Bibliography_of_E._T._Whittaker
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 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
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
French-American mathematician
research 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
German mathematician (1886–1982)
doctorate in 1910. His dissertation was titled On the theory of automorphic functions (German: Theorie der automorphen Funktionen). He began working as
Ludwig_Bieberbach
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
English mathematician (born 1947)
1999. Jacques Hadamard: Non-Euclidean Geometry in the Theory of Automorphic Functions (with Abe Shenitzer), translated Abe Shenitzer with an introductory
Jeremy_Gray_(mathematician)
Shimura, Gorō (1971). Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. ISBN 978-0-691-08092-5
Hecke_algebra_of_a_pair
Field extension that is not algebraic
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Tokyo:
Transcendental_extension
French mathematician, physicist and engineer (1854–1912)
field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number
Henri_Poincaré
associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice. Suppose that L is a positive definite lattice. The Siegel
Siegel_theta_series
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)
Function whose domain is the positive integers
theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range
Arithmetic_function
Mathematical function
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
Dedekind_eta_function
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
Japanese mathematician
especially analytic number theory, multiple trigonometric function theory, zeta functions and automorphic forms. He is currently a professor emeritus at Tokyo
Nobushige_Kurokawa
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
Plane algebraic curve
Serge Lang, Elliptic Functions, Addison-Wesley, 1973 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1972 OEIS
Classical_modular_curve
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
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
Mathematical map for transforming elliptic modular forms
Hidehisa (1967), "On the algebraic curves uniformized by arithmetical automorphic functions", Annals of Mathematics, Second Series, 86: 449–460, doi:10.2307/1970610
Doi–Naganuma_lifting
Mathematical function
In mathematics, Hooley's delta function, also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle n} in [ u
Hooley's_delta_function
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
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AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Biblical
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Male
Celtic
, great justiciary, or functionary.
Male
Egyptian
, a high Egyptian functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
AUTOMORPHIC FUNCTION
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