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Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations into $$\mathrm{GL}_2(\mathbb{Z}_{\ell} \otimes \mathcal{O}_F)$$ with exceptional set $\ell \mid N$. The standard way to see this is that it is the tensor product of the first $\ell$-adic cohomology of the modular curve of level $N$ with $F$ over the Hecke algebra, where we view $F$ as an algebra over the Hecke algebra via $f$.

I read that there is a construction that gives not only the representation but an abelian variety with $\ell$-adic representation equal to that. Could someone please give a readable reference for this? Thanks.

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Dear David, This construction is due to Shimura, and can be found in his book. (Incidentally, it's not necessary to require the coefficients of $f$ to be totally real). In short, one does as you wrote, but works with the Jacobian of the modular curve rather than just its $\ell$-adic Tate module (which is what the $\ell$-adic cohomology of the modular curve is), i.e. if $X$ is the relevant modular curve and $\lambda:\mathbb T\to \mathcal O_F$ is the system of eigenvalues corresponding to $f$, then you form the abelian variety $A_f: = $Jac$(X)\otimes_{\mathbb T}\mathcal O_F$. Regards, Matthew – Emerton Aug 2 '12 at 2:40
I understand that in the analytic category, but is there actually a notion of tensor product in the algebraic category? – David Corwin Aug 2 '12 at 14:56
Dear David, Yes, there is. Regards, Matthew – Emerton Aug 4 '12 at 3:41
But I remember being very confused about this precise point as a graduate student until Brian Conrad explained it to me. – JSE Aug 5 '12 at 2:13

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