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automorphic

Spanish translation: funciones (o fórmulas) automorfas..

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GLOSSARY ENTRY (DERIVED FROM QUESTION BELOW)
English term or phrase:automorphic
Spanish translation:funciones (o fórmulas) automorfas..
Entered by: Gabriela Rodriguez
Options:
- Contribute to this entry
- Include in personal glossary

18:15 Jun 15, 2005
English to Spanish translations [PRO]
Social Sciences - Mathematics & Statistics
English term or phrase: automorphic
Includes instruction in algebraic structures, quadratic and automorphic forms, combinatorics, linear algebra, and algebraic geometry
InGoodSpanish
Local time: 03:05
funciones (o fórmulas) automorfas.
Explanation:
Suerte!!!!!!!

--------------------------------------------------
Note added at 3 mins (2005-06-15 18:19:52 GMT)
--------------------------------------------------

Biografía Matemáticos:David Hilbert. Versión para imprimir
Como vemos, una ojeada superficial a la actividad matemática de Hilbert en ...
la parametrización de curvas algebraicas por medio de funciones automorfas. ...
www.divulgamat.net/weborriak/Historia/ MateOspetsuak/Inprimaketak/Hilbert.asp - 42k

--------------------------------------------------
Note added at 4 mins (2005-06-15 18:20:10 GMT)
--------------------------------------------------

DF] Matemáticas
Formato de archivo: PDF/Adobe Acrobat - Versión en HTML
Área de conocimiento: Análisis Matemático (015) y Matemática Aplicada (595). ...
Teoría espectral de formas automorfas. Problemas de puntos del retículo. ...
www.uam.es/estudios/doctorado/ programas/CIENCIAS/MATEMATICAS.pdf

--------------------------------------------------
Note added at 4 mins (2005-06-15 18:20:48 GMT)
--------------------------------------------------

automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. It is in terms of a Lie group G, to generalise the groups SL2(R) or PSL2(R) of modular forms, and a discrete group Γ in G, to generalise the modular group, or one of its congruence subgroups. The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j 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 j is derived from a Jacobian matrix, by means of the chain rule.

In the general setting, then, an automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

1. to transform under translation by elements γ of Γ according to the given automorphy factor j;
2. to be an eigenfunction of certain Casimir operators on G; and
3. to satisfy some conditions on growth at infinity.

It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for γ in Γ. In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to \'twist\' them. The Casimir operator condition says that some Laplacians have F as eigenfunction; this ensures that F 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 G/Γ is not compact but has cusps.

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 Γ a Fuchsian group had already received attention before 1900. The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that, though a full theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions. 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 Pyatetskii-Shapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Langlands showed how (in generality, many cases being known) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. He also produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would be the \'continuous spectrum\' for this problem, leaving the cusp form or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Ramanujan, as the heart of the matter.

The subsequent notion of automorphic representation has proved of great technical value for dealing with G an algebraic group, treated as an adelic algebraic group. It does not actually completely include the automorphic form idea introduced above, in that the adele approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representation that is an infinite tensor product of representations of p-adic groups, with specific enveloping algebra representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.
http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey...
Selected response from:

Gabriela Rodriguez
Argentina
Local time: 03:05
Grading comment
2 KudoZ points were awarded for this answer

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Summary of answers provided
3 +5automórfico(a)xxxOso
4 +1funciones (o fórmulas) automorfas.
Gabriela Rodriguez


  

Answers


1 min   confidence: Answerer confidence 3/5Answerer confidence 3/5 peer agreement (net): +5
automórfico(a)


Explanation:
Un numero se dice automorfico si su cuadrado termina en los
mismos digitos que el numero original, por ejemplo, 76 2 ¡ 5776. ...
www.decom-uv.cl/~rsalas/Ramos/ Mat-258/Pautas/Pauta_1_Mat-258.pdf

--------------------------------------------------
Note added at 2005-06-15 18:19:05 (GMT)
--------------------------------------------------

\"... Encuentre los números automórficos en el rango del 1 al 100. Un número
automórfico es aquel que reaparece al final de su cuadrado. ... \"
members.fortunecity.es/irmaprado/tarea12.htm

Buena suerte y saludos del Oso ¶:^)

xxxOso
Native speaker of: Native in SpanishSpanish
PRO pts in category: 4

Peer comments on this answer (and responses from the answerer)
agree  Maria Milagros Del Cid
0 min
  -> Muchas gracias, María ¡Saludos! ¶:^)

agree  Marc Figueras
2 mins
  -> Muchas gracias, Marc ¶:^)

agree  Ernesto de Lara
16 mins
  -> Muchas gracias, Ernesto ¶:^)

agree  Hebe Martorella
1 hr
  -> Hola Hebe, muchísimas gracias ¶:^)

agree  Dark Angel
1 hr
  -> Muchas gracias, Dark Angel (cool name!) ¶:^)
Login to enter a peer comment (or grade)

3 mins   confidence: Answerer confidence 4/5Answerer confidence 4/5 peer agreement (net): +1
funciones (o fórmulas) automorfas.


Explanation:
Suerte!!!!!!!

--------------------------------------------------
Note added at 3 mins (2005-06-15 18:19:52 GMT)
--------------------------------------------------

Biografía Matemáticos:David Hilbert. Versión para imprimir
Como vemos, una ojeada superficial a la actividad matemática de Hilbert en ...
la parametrización de curvas algebraicas por medio de funciones automorfas. ...
www.divulgamat.net/weborriak/Historia/ MateOspetsuak/Inprimaketak/Hilbert.asp - 42k

--------------------------------------------------
Note added at 4 mins (2005-06-15 18:20:10 GMT)
--------------------------------------------------

DF] Matemáticas
Formato de archivo: PDF/Adobe Acrobat - Versión en HTML
Área de conocimiento: Análisis Matemático (015) y Matemática Aplicada (595). ...
Teoría espectral de formas automorfas. Problemas de puntos del retículo. ...
www.uam.es/estudios/doctorado/ programas/CIENCIAS/MATEMATICAS.pdf

--------------------------------------------------
Note added at 4 mins (2005-06-15 18:20:48 GMT)
--------------------------------------------------

automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. It is in terms of a Lie group G, to generalise the groups SL2(R) or PSL2(R) of modular forms, and a discrete group Γ in G, to generalise the modular group, or one of its congruence subgroups. The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j 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 j is derived from a Jacobian matrix, by means of the chain rule.

In the general setting, then, an automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

1. to transform under translation by elements γ of Γ according to the given automorphy factor j;
2. to be an eigenfunction of certain Casimir operators on G; and
3. to satisfy some conditions on growth at infinity.

It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for γ in Γ. In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to \'twist\' them. The Casimir operator condition says that some Laplacians have F as eigenfunction; this ensures that F 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 G/Γ is not compact but has cusps.

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 Γ a Fuchsian group had already received attention before 1900. The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that, though a full theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions. 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 Pyatetskii-Shapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Langlands showed how (in generality, many cases being known) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. He also produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would be the \'continuous spectrum\' for this problem, leaving the cusp form or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Ramanujan, as the heart of the matter.

The subsequent notion of automorphic representation has proved of great technical value for dealing with G an algebraic group, treated as an adelic algebraic group. It does not actually completely include the automorphic form idea introduced above, in that the adele approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representation that is an infinite tensor product of representations of p-adic groups, with specific enveloping algebra representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.
http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey...

Gabriela Rodriguez
Argentina
Local time: 03:05
Native speaker of: Native in SpanishSpanish
PRO pts in category: 26

Peer comments on this answer (and responses from the answerer)
agree  Marc Figueras: Sí, también "automorfas"... de hecho a mi me gusta más "automorfa" que "automórfica".
3 mins
  -> Muchísimas gracias Marc y un gran saludo!!!!!!!!
Login to enter a peer comment (or grade)




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