1101:, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a discrete infinite group of linear fractional transformations. Automorphic functions then generalize both
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For fifteen days I strove to prove that there could not be any functions like those I have since called
Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary
1117:
to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of
Fuchsian functions, those which come from the
784:
may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when
1175:
979:. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by
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Before this very general setting was proposed (around 1960), there had already been substantial developments of automorphic forms other than modular forms. The case of Î a
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has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where
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939:; because of their symmetry properties. Therefore, in simpler terms, a general function which analyzes the invariance of a structure with respect to its prime
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is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of
1097:'s first discoveries in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician
847:
Examples of automorphic forms in an explicit unabstracted state are difficult to obtain, though some have directly analytical properties:
1269:
1250:
258:
233:. In particular, the connections between multiple attempts at definition are unclear; see talk "Definition section is confusing".
136:; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the
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1000:
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1226:
832:. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of
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1312:
959:(also called Hilbert-Blumenthal forms) were proposed not long after that, though a full theory was long in coming. The
1221:
844:. Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure.
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or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since
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992:
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The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with
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are here in effect put on the same level as the
Casimir operators; which is natural from the point of view of
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1081:, though not so obviously for the number theory. It is this concept that is basic to the formulation of the
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919:). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore
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extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the
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could be applied to the calculation of dimensions of automorphic forms; this is a kind of
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738:Ï acting on the components to 'twist' them. The Casimir operator condition says that some
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of the topological group. Automorphic forms are a generalization of the idea of
734:. In the vector-valued case the specification can involve a finite-dimensional
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check on the validity of the notion. He also produced the general theory of
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1258:, "Automorphic Forms and Representations", 1998, Cambridge University Press
1307:
30:
983:, in the years around 1960, in creating such a theory. The theory of the
1057:, an automorphic representation is a representation that is an infinite
1007:
terms would be the 'continuous spectrum' for this problem, leaving the
885:
121:
911:, which are invariant with respect to a generalized analogue of their
987:, as applied by others, showed the considerable depth of the theory.
144:
at once. From this point of view, an automorphic form over the group
113:
618:, in the vector-valued case), subject to three kinds of conditions:
204:, automorphic forms play an important role in modern number theory.
1121:; I had only to write out the results, which took but a few hours.
991:
showed how (in generality, many particular cases being known) the
897:
29:
800:
A more straightforward but technically advanced definition using
1275:
This article incorporates material from Jules Henri
Poincaré on
903:
As a general principle, automorphic forms can be thought of as
27:
Type of generalization of periodic functions in
Euclidean space
211:
1264:
Stephen
Gelbart (1975), "Automorphic forms on Adele groups",
955:
had already received attention before 1900 (see below). The
1089:
Poincaré on discovery and his work on automorphic functions
1073:(s). One way to express the shift in emphasis is that the
186:) and satisfies certain smoothness and growth conditions.
112:
are holomorphic automorphic forms defined over the groups
192:
first discovered automorphic forms as generalizations of
1112:
Poincaré explains how he discovered
Fuchsian functions:
931:
structure. In the simplest sense, automorphic forms are
1045:
approach is a way of dealing with the whole family of
673:
to satisfy a "moderate growth" asymptotic condition a
140:
approach as a way of dealing with the whole family of
1237:
Spectral
Methods of Automorphic Forms, Second Edition
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106:in Euclidean space to general topological groups.
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1245:), American Mathematical Society, Providence, RI
1281:Creative Commons Attribution/Share-Alike License
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765:The formulation requires the general notion of
611:(with values in some fixed finite-dimensional
8:
239:. There might be a discussion about this on
38:is an automorphic form in the complex plane.
1151:, a book by H. Jacquet and Robert Langlands
1053:space for a quotient of the adelic form of
622:to transform under translation by elements
820:. Herein, the analytical structure of its
1176:"Automorphic Forms: A Brief Introduction"
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259:Learn how and when to remove this message
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824:allows for generalizations with various
463:{\displaystyle f(g\cdot x)=j_{g}(x)f(x)}
1166:
917:irreducible fundamental representation
128:with the discrete subgroup being the
7:
746:as eigenfunction; this ensures that
681:It is the first of these that makes
310:acts on a complex-analytic manifold
971:, arose naturally from considering
374:to the complex numbers. A function
165:, is a complex-valued function on
727:{\displaystyle \gamma \in \Gamma }
721:
688:, that is, satisfy an interesting
641:{\displaystyle \gamma \in \Gamma }
635:
603:An automorphic form is a function
83:
54:is a well-behaved function from a
25:
580:is an automorphic form for which
1306:
1294:
216:
95:{\displaystyle \Gamma \subset G}
1242:Graduate Studies in Mathematics
1003:, which corresponds to what in
178:) that is left invariant under
69:) which is invariant under the
1279:, which is licensed under the
1015:, as the heart of the matter.
869:- Specific generalizations of
828:properties; and the resultant
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161:and an algebraic number field
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772:for Î, which is a type of 1-
1222:Encyclopedia of Mathematics
1019:Automorphic representations
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1148:Automorphic Forms on GL(2)
1022:
892:which is invariant on its
350:also acts on the space of
157:), for an algebraic group
18:Automorphic representation
854:(which is a prototypical
536:for the automorphic form
288:complex-analytic manifold
1069:representations for the
502:{\displaystyle j_{g}(x)}
398:if the following holds:
1239:, (2002) (Volume 53 in
1025:Cuspidal representation
923:), constructed by some
648:according to the given
1299:Quotations related to
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1061:of representations of
1039:adelic algebraic group
981:Ilya Piatetski-Shapiro
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1119:hypergeometric series
985:Selberg trace formula
957:Hilbert modular forms
875:class field-theoretic
871:Dirichlet L-functions
842:fundamental structure
818:Artin reciprocity law
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202:Langlands conjectures
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36:Dedekind eta-function
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1315:at Wikimedia Commons
1174:Friedberg, Solomon.
1137:Automorphic function
1083:Langlands philosophy
1047:congruence subgroups
993:RiemannâRoch theorem
961:Siegel modular forms
808:to their underlying
767:factor of automorphy
736:group representation
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229:confusing or unclear
142:congruence subgroups
134:congruence subgroups
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1079:functional analysis
1049:at once. Inside an
1013:Srinivasa Ramanujan
935:defined on general
909:abstract structures
776:in the language of
690:functional equation
237:clarify the section
1217:"Automorphic Form"
1132:Automorphic factor
1107:elliptic functions
1067:enveloping algebra
915:(or an abstracted
905:analytic functions
802:class field theory
793:, by means of the
789:is derived from a
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290:. Suppose a group
198:elliptic functions
104:periodic functions
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1334:Automorphic forms
1313:Automorphic forms
1311:Media related to
1001:Eisenstein series
894:ideal class group
882:harmonic analytic
852:Eisenstein series
830:Langlands program
826:algebro-geometric
814:idele class group
664:Casimir operators
600:is the identity.
593:{\displaystyle j}
569:{\displaystyle j}
549:{\displaystyle f}
522:{\displaystyle G}
387:{\displaystyle f}
367:{\displaystyle X}
343:{\displaystyle G}
323:{\displaystyle X}
303:{\displaystyle G}
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75:discrete subgroup
56:topological group
44:harmonic analysis
16:(Redirected from
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1301:Automorphic form
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1181:. Archived from
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1065:, with specific
1037:, treated as an
989:Robert Langlands
969:symplectic group
880:- Generally any
860:field extensions
780:. The values of
778:group cohomology
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1142:Maass cusp form
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1035:algebraic group
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1005:spectral theory
977:theta functions
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927:analogue on an
921:elliptic curves
858:) over certain
791:Jacobian matrix
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1233:Henryk Iwaniec
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194:trigonometric
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48:number theory
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1303:at Wikiquote
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1190:. Retrieved
1183:the original
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964:
963:, for which
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884:object as a
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67:vector space
65:(or complex
58:
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1256:Daniel Bump
1192:10 February
1155:Jacobi form
929:automorphic
913:prime ideal
686:automorphic
662:of certain
273:mathematics
1329:Lie groups
1323:Categories
1277:PlanetMath
1205:References
1023:See also:
937:Lie groups
822:L-function
816:under the
795:chain rule
754:/Î is not
740:Laplacians
231:to readers
208:Definition
1227:EMS Press
1215:(2001) ,
1009:cusp form
877:objects.
840:of their
722:Γ
719:∈
716:γ
692:relating
658:to be an
636:Γ
633:∈
630:γ
418:⋅
249:July 2023
87:⊂
84:Γ
1126:See also
1095:Poincaré
997:post hoc
758:but has
330:. Then,
190:Poincaré
1093:One of
947:History
886:functor
774:cocycle
756:compact
700:) with
227:may be
122:PSL(2,
61:to the
1268:
1249:
1043:adelic
850:- The
708:) for
473:where
284:acting
138:adelic
114:SL(2,
71:action
1186:(PDF)
1179:(PDF)
1161:Notes
967:is a
898:idele
888:over
760:cusps
742:have
670:; and
576:. An
354:from
286:on a
281:group
73:of a
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1266:ISBN
1247:ISBN
1194:2014
1105:and
975:and
896:(or
532:The
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873:as
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271:In
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126:)
124:R
118:)
116:R
90:G
59:G
20:)
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