Is $ pimapsto(smapsto L(s,pi)) $ bijective?
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Let $ pi $ be an automorphic representation of $ operatorname{GL_{n}}(mathbb{A}_{mathbb{Q}}) $ and $ L(s,pi) $ the associated L-function. Is the map $ pimapsto L(s,pi) $ bijective ? In other words, is the knowledge of $ L(s,pi) $ exactly equivalent to the knowledge of $ pi $ itself ?
number-theory automorphic-forms l-functions
asked Dec 5 at 12:14
Sylvain Julien
1,101918
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Let $ pi $ be an automorphic representation of $ operatorname{GL_{n}}(mathbb{A}_{mathbb{Q}}) $ and $ L(s,pi) $ the associated L-function. Is the map $ pimapsto L(s,pi) $ bijective ? In other words, is the knowledge of $ L(s,pi) $ exactly equivalent to the knowledge of $ pi $ itself ?
number-theory automorphic-forms l-functions
asked Dec 5 at 12:14
Sylvain Julien
1,101918
How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
1
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43
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down vote
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Let $ pi $ be an automorphic representation of $ operatorname{GL_{n}}(mathbb{A}_{mathbb{Q}}) $ and $ L(s,pi) $ the associated L-function. Is the map $ pimapsto L(s,pi) $ bijective ? In other words, is the knowledge of $ L(s,pi) $ exactly equivalent to the knowledge of $ pi $ itself ?
number-theory automorphic-forms l-functions
asked Dec 5 at 12:14
Sylvain Julien
1,101918
Let $ pi $ be an automorphic representation of $ operatorname{GL_{n}}(mathbb{A}_{mathbb{Q}}) $ and $ L(s,pi) $ the associated L-function. Is the map $ pimapsto L(s,pi) $ bijective ? In other words, is the knowledge of $ L(s,pi) $ exactly equivalent to the knowledge of $ pi $ itself ?
number-theory automorphic-forms l-functions
number-theory automorphic-forms l-functions
asked Dec 5 at 12:14
Sylvain Julien
1,101918
asked Dec 5 at 12:14
Sylvain Julien
1,101918
asked Dec 5 at 12:14
Sylvain Julien
1,101918
asked Dec 5 at 12:14
Sylvain Julien
1,101918
asked Dec 5 at 12:14
Sylvain Julien
1,101918
1,101918
How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
1
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43
add a comment |
How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
1
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43
How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
1
1
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43
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How an irreducible automorphic representation is defined by the isomorphism classes (Satake parameters - local L-functions) of its local representations is in every text en.wikipedia.org/wiki/Multiplicity-one_theorem
– reuns
Dec 5 at 12:21
In this setting an automorphic representation isn't a function $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ but the abstract linear maps acting on the vector space generated by the (right) translates of $f$. So $f(.g)$ isn't the same function as $f$ but it is the same representation.
– reuns
Dec 5 at 12:33
Call $f : GL_n(Q) setminus GL_n(A_Q) to mathbb{C}$ an automorphic form and $pi(f)$ its automorphic representation (so $pi(f)(g)$ is the linear map sending $x mapsto u(x)=sum_j a_j f(xy_j)in V(f)$ to $x mapsto u(xg) in V(f)$). If $pi(f_1),pi(f_2)$ are two irreducible automorphic representations then $pi(f_1+f_2)$ is an automorphic representation. What is its decomposition in direct sum of irreducible automorphic representations ? What is its L-function ?
– reuns
Dec 5 at 13:47
1
Look up "strong multiplicity one".
– Kimball
Dec 8 at 16:43