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nt.quantity principle – Automorphic factorization of Dedekind zeta capabilities Answer

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nt.quantity principle – Automorphic factorization of Dedekind zeta capabilities

It is well-known that for abelian quantity fields, the factorization of its Dedekind zeta perform goes love this:

$$zeta_K(s)=zeta(s)prod_{chi neq 1} L(s,chi)$$

with the Dirichlet characters discrete and raw.

If $Okay$ is non-abelian however Galois, we as a substitute have (by Aramata-Brauer):

$$zeta_K(s)=zeta(s)prod_{rho neq 1} L(s,rho)$$

with the representations non-trivial and irreducible, though on this illustration we do not know unconditionally that that is a factorization of irreducible L-functions. In the non-Galois illustration, elements energy emerge with capricious integer powers. But let’s neglect about that for a significance.

In this respond, Kevin Dong mentions an categorical factorization of the zeta of $mathbb{Z}[sqrt[3]{2}]$ when it comes to modular kinds. It may be very good however not fairly suprising: Artin L-functions are anticipated (and identified in some instances) to at all times breathe automorphic.

  • I’m on the proof for $mathbb{Z}[sqrt[3]{2}]$ (I have not been capable of finding a reference for it), and another reference for identified illustration of such a factorization (that is, not a factorization on phrases of Artin L-function, however of automorphic ones).

  • I’d too need to know what the conjectures are (on the automorphic aspect) for what the factorization seems to be love for an capricious non-Galois Dedekind zeta perform.

Any different data round these points energy breathe of ameliorate, however not usually about Langlands or the Artin surmise.

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