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GH from MO
GH from MO
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I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).

I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can sometimes deduce that the $L$-function is itself automorphic (not just pretends to be).

Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).

I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).

I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can deduce that the $L$-function is itself automorphic (not just pretends to be).

Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).

I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).

I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can sometimes deduce that the $L$-function is itself automorphic (not just pretends to be).

Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).

Source Link
GH from MO
GH from MO
  • 105.4k
  • 8
  • 293
  • 398

I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).

I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can deduce that the $L$-function is itself automorphic (not just pretends to be).

Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).