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Edited after Will Sawin's comment:

Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product $\otimes$ such that $\forall p\in\mathbb{P}, \ \ a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ where $a_{n}(H)$ is the $n$-th Dirichlet coefficient of $H$, that is $H(s)=\displaystyle{\sum_{n\gt 0}\dfrac{a_{n}(H)}{n^s}}$ whenever $\Re(s)\gt 1$. This tensor product corresponds, on the automorphic side, to Rankin-Selberg convolution. For the sake of simplicity, the term of 'L-function' will be used to mean any element of $\mathcal{M}$.

Let's define the automorphism group of $\mathcal{M}$ as the group, under composition, of the bijective maps $\Phi$ from $\mathcal{M}$ to itself such that the following properties are simultaneously fulfilled:

A) $\Phi$ maps a primitive L-function to a primitive L-function

B) $\forall (F,G)\in\mathcal{M}^{2}, \ \ \Phi(F\odot G)=\Phi(F)\odot\Phi(G)$ where $\odot\in\{\times, \otimes\}$

Such an automorphism of $\mathcal{M}$ preserves the degree of any L-functions, that is $d_{\Phi(F)}=d_{F}$.

Assuming that for any two L-functions $F$ and $G$, $d_{F\otimes G}=d_{F}.d_{G}$, let's now associate to an L-function $H$ a complex manifold $X_{H}$ of dimension $d_{H}$ such that for any two L-functions $F$ and $G$, $X_{F.G}=X_{F}\oplus X_{G}$ and $X_{F\otimes G}=X_{F}\otimes X_{G}$ so that $F=G\Leftrightarrow X_{F}=X_{G}$. We shall denote the set of all $X_{F}$ where $F$ runs over $\mathcal{M}$ by $\mathcal{M}'$ and any element of $\mathcal{M}'$ will be called an L-manifold.

The automorphism group of $\mathcal{M}'$ is defined in a similar fashion as for the one of $\mathcal{M}$ so that these two groups are isomorphic.

Let's now define the notion of abstract Galois group $\operatorname{Gal}(A/B)$ as the set of all automorphisms of $A$ that preserve $B$ pointwise, and let's associate to any L-function $F$ its ' canonical' representation $(\rho_{F}, V_{F})$ such that the following properties are simultaneously fulfilled:

C) there exists an algebraic number field $K_{F}$ the absolute Galois group of which, denoted by $G_{K_{F}}$, is isomorphic to both $\operatorname{Gal}(\mathcal{M}/<F>)$ and $\operatorname{Gal}(\mathcal{M}'/<X_F>)$, and is such that $\rho_{F}$ is a group homomorphism from $G_{K_{F}}$ to $\operatorname{GL}_{d_{F}}(\mathbb{C})=Aut(V_{F})$ where $<F>=\{\bigodot_{k=0}^{m}F, \odot\in\{\times,\otimes\}, m\in\mathbb{N}_{0}\}$, $<X_{F}>$ is defined in a similar way, and so that $X_{F}$ is locally isomorphic to $V_{F}$.

Assuming $F$ is the L-function attached to an automorphic representation of $\operatorname{GL}_{d_{F}}(\mathbb{A}_{K_{F}})$ where $\mathbb{A}_{K_{F}}$ is the adele ring of $K_{F}$ we require that the considered representation $(\rho_{F}, V_{F})$ is faithful, and that it is irreducible if and only if $F$ is primitive. We also require that $V_{F.G}=V_{F}\oplus V_{G}$ and that $V_{F\otimes G}=V_{F}\otimes V_{G}$.

D) $F(s)=L(s,\rho_{F})$

My questions are:

1) does every L-manifold give rise naturally to a motive?

2) If so, is every L-function motivic? Can one say that any motivic L-function arises from a Galois representation and conversely?

Many thanks in advance.

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closed as unclear what you're asking by Will Sawin, Dan Petersen, Stefan Kohl, Lucia, Dima Pasechnik Apr 26 '15 at 23:08

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  • $\begingroup$ I think this is a very cool question. Could you explain, or give a reference, to the notion of the L-functions "canonical" representation? Part of this includes the definition of the field $K_F$ as well. $\endgroup$ – WSL Apr 26 '15 at 12:22
  • $\begingroup$ The considered field is defined up to isomorphism by its absolute Galois group, as follows from Neukirch-Ikeda-Uchida theorem. $\endgroup$ – Sylvain JULIEN Apr 26 '15 at 12:38
  • $\begingroup$ Okay thanks, but I still would like to know what the canonical representation is. Also, is there a method for attaching to an L-function $H$ an abstract variety, $X_H$, you are refering to? $\endgroup$ – WSL Apr 26 '15 at 13:33
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    $\begingroup$ There's no such thing as the tensor product of abstract varieties, which you use in the definition of $X_F$. Your second question doesn't really make sense - each variety has a motive attached to it, but you can't interpret a variety as a motive - and your third question seems like a restatement of one of the Langlands conjectures. $\endgroup$ – Will Sawin Apr 26 '15 at 22:04
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    $\begingroup$ Maybe. If you want to define something significant in this field, you should almost certainly try to construct it rather than characterize it. It is very hard to find good constructions of things related to automorphic forms/Galois representations, so characterizations are not so useful unless you know how to construct something satisfying them. It would also clarify what you mean, which is not very clear right now. Note that there is also no tensor product of abstract topological spaces. $\endgroup$ – Will Sawin Apr 26 '15 at 22:32