Galois representations attached to modular forms of weight one Why are the Galois representations attached to weight one eigenforms by Serre-Deligne complex representations?
In weight $k\geq 2$, the Galois representations constructed by Eichler-Shimura and Deligne are all $\ell$-adic representations. 
The weight one case seems to stand on its own. Is there an a priori reason why one should expect this difference between $k=1$ and $k\geq 2$? What is special about weight one forms?
 A: In each case (weight $1$ or weight $k \geq 2$), there is a number field $E$ (the number field generated by the prime-to-the-level Hecke eigenvalues of $f$) and
a compatible family of $\lambda$-adic reps. $\rho_{\lambda}$ attached to $f$, where compatible means
that if $p$ doesn't divide the level, and is prime-to-$\lambda$,
then $\rho_{\lambda}$ is unram. at $p$, and its trace on $Frob_p$ 
is equal to the $p$th Hecke eigenvalue (an element of $E$ which doesn't depend
on $\lambda$).
In the case when $k = 1$, one has the additional fact that all the $\rho_{\lambda}$ actually have finite image, so (using Cebotarev) are defined over $E$ (not just $E_{\lambda}$), and hence (by the compatibility) actually coincide.
By embedding $E$ in $\mathbb C$, you can of course think of them as complex reps. as well.  (Or, if you like, in this case there is no need to remember that the Hecke eigenvalues all lie in a common number field $E$; you can just think of them as complex numbers.  But this view-point obscures the unity between the $k =1$ and $k \geq 2$ situations.)
Why is there this difference in behaviour?
Well, these reps. are constructed out of etale cohomology of some motive
(the different $\lambda$ coming from the different choices for $\ell$ in the
construction of $\ell$-adic cohom.).
E.g. in the case $k = 2$ they come from Tate modules (or duals of Tate modules,
if you want to think cohomologically rather than homologically) of abelian varieties.
In general, this cohomology is in dimension $k - 1$.
If $k \geq 2$, then $k - 1 \geq 1,$ and there is no "etale cohomology with 
$\mathbb Q$ coefficients" that underlies the different $\ell$-adic etale cohomologies.  There are relations between the different $\ell$-adic cohomologies, and this is where the compatibility comes from.  But the different $\lambda$-adic reps. are not in any sense the same; they are just related via compatibility.
But when $k = 1,$ we have $k - 1 = 0$, and we can define etale $H^0$ with $\mathbb Q$-coeffs.
of a variety: it is just the $\mathbb Q$-v.s. generated by the set of geometrically connected components (or maybe the dual, if you want cohom. rather than homology).  This set of components is finite, and so the Galois action factors through a finite quotient.  So precisely in this case can we find a common (finite image) rep'n underlying the different $\lambda$-adic reps. 
