3 added 1 characters in body

Regarding (1), from the point of view of Galois representations, the point is that continuous Weil group representations on a complex vector space, by their nature, have finite image on inertia.

On the other hand, while a continuous $\ell$-adic Galois representation of $G_{\mathbb Q_p}$ (with $\ell \neq p$ of course) must have finite image on wild inertia, it can have infinite image on tame inertia. The formalism of Weil--Deligne representations extracts out this possibly infinite image, and encodes it as a nilpotent operator (something that is algebraic, and doesn't refer to the $\ell$-adic topology, and hence has a chance to be independent of $\ell$).

As for (2): Representations of the Weil group are essentially the same thing as representations of $G_{\mathbb Q}$ which, when restricted to some open subgroup, become abelian. Thus (as one example) if $E$ is an elliptic curve over $\mathbb Q$ that is not CM, its $\ell$-adic Tate module cannot be explained by a representation of the Weil group (or any simple modification thereof). Thus neither can the weight 2 modular form to which it corresponds.

In summary: the difference between the global and local situations is that an $\ell$-adic representation of $G_{\mathbb Q_p}$ (or of $G_E$ for any $p$-adic local field) becomes, after a finite base-change to kill of off the action of wild inertia, a tamely ramified representation, which can then be described by two matrices, the image of a lift of Frobenius and the image of a generator of tame inertia, satisfying a simple commutation relation.

On the other hand, global Galois representations arising from $\ell$-adic cohomology of varieties over number fields are much more profoundly non-abelian.

Added: Let me also address the question about a product of $W_{F_v}'$. Again, it is simplest to think in terms of Galois representations (which roughly correspond to motives, which, one hopes, roughly correspond to automorphic forms).

So one can reinterpret the question as asking: is giving a representation of $G_F$ (for a number field $F$) the same as giving representations of each $G_{F_v}$ (as $v$ ranges over the places of $F$). Certainly, by Cebotarev, the restriction of the global representation to the local Galois groups will determine it; but it will overdetermine it; so giving a collection of local representations, it is unlikely that they will combine into a global one. ($G_F$ is very far from being the free product of the $G_{F_v}$, as Cebotarev shows.)

To say something on the automorphic side, imagine writing down a random degree 2 Euler product. You can match this with a formal $q$-expansion, which will be a Hecke eigenform, by taking Mellin transforms, and with a representation of $GL_2(\mathbb A_F)$, by writing down a corresponding tensor product of unramified representations of the various $G_{F_v}$. But what chance is there that this object is an automorphic representation? What chance is there that your random formal Hecke eigenform is actually a modular form? What chance is there that your random Euler product is actually an automorphic $L$-function? Basically none.

You have left out some vital global glue, the same glue which describes the interrelations of all the $G_{F_v}$ inside $G_F$. Teasing out the nature of this glue is at the heart of proving the conjectured relationship between automorphic forms and motives; its mysterious nature is what makes the theories of automorphic forms, and of Galois representations, so challenging.

2 added 1819 characters in body; edited body
which can then be described by two matrices, the image of a lift of Frobenius and the image of a generator of tame inertia, satisfying a simple computation commutation relation.

On the other hand, global Galois representations arising from $\ell$-adic cohomology of varieties over number fields are much more profoundly non-abelian.

Added: Let me also address the question about a product of $W_{F_v}'$. Again, it is simplest to think in terms of Galois representations (which roughly correspond to motives,which, one hopes, roughly correspond to automorphic forms).

So one can reinterpret the question as asking: is giving a representation of $G_F$ (for a number field $F$) the same as giving representations of each $G_{F_v}$ (as $v$ ranges over the places of $F$). Certainly, by Cebotarev, the restriction of the global representationto the local Galois groups will determine it; but it will overdetermine it; so giving a collection of local representations, it is unlikely that they will combine into a global one. ($G_F$ is very far from being the free product of the $G_{F_v}$, as Cebotarev shows.)

To say something on the automorphic side, imagine writing down a random degree 2 Euler product. You can match this with a formal $q$-expansion, which will be a Hecke eigenform, by taking Mellin transforms,and with a representation of $GL_2(\mathbb A_F)$, by writing down a corresponding tensor product of unramified representations of the various $G_{F_v}$. But what chance is there that this object is an automorphic representation? What chance is there that your random formal Hecke eigenform is actually a modular form? What chance is there that your random Euler product is actually an automorphic $L$-function? Basically none.

You have left out some vital global glue, the same glue which describes the interrelations of all the $G_{F_v}$ inside $G_F$. Teasing out the nature of this glue is at the heart of proving the conjectured relationship between automorphic forms and motives; its mysterious nature is what makes the theories of automorphic forms, and of Galois representations, so challenging.

1

Regarding (1), from the point of view of Galois representations, the point is that continuous Weil group representations on a complex vector space, by their nature, have finite image on inertia.

On the other hand, while a continuous $\ell$-adic Galois representation of $G_{\mathbb Q_p}$ (with $\ell \neq p$ of course) must have finite image on wild inertia, it can have infinite image on tame inertia. The formalism of Weil--Deligne representations extracts out this possibly infinite image, and encodes it as a nilpotent operator (something that is algebraic, and doesn't refer to the $\ell$-adic topology, and hence has a chance to be independent of $\ell$).

As for (2): Representations of the Weil group are essentially the same thing as representations of $G_{\mathbb Q}$ which, when restricted to some open subgroup, become abelian. Thus (as one example) if $E$ is an elliptic curve over $\mathbb Q$ that is not CM, its $\ell$-adic Tate module cannot be explained by a representation of the Weil group (or any simple modification thereof). Thus neither can the weight 2 modular form to which it corresponds.

In summary: the difference between the global and local situations is that an $\ell$-adic representation of $G_{\mathbb Q_p}$ (or of $G_E$ for any $p$-adic local field) becomes, after a finite base-change to kill of the action of wild inertia, a tamely ramified representation, which can then be described by two matrices, the image of a lift of Frobenius and the image of a generator of tame inertia, satisfying a simple computation relation.

On the other hand, global Galois representations arising from $\ell$-adic cohomology of varieties over number fields are much more profoundly non-abelian.