If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in Shimura's book, but was wondering if anyone knows of a different reference (perhaps one that is a bit more readable…).
Thanks.
Best Answer
Have a look at Section 6.6 of Diamond and Shurman, A First Course in Modular Forms:
As an aside, the theorem states a bit more than you have said: for instance, when the field of Fourier coefficients is $\mathbb{Q}$, you are just asserting the existence of an elliptic curve $E_{/\mathbb{Q}}$ with $\operatorname{End}_{\mathbb{Q}}(E) \otimes_{\mathbb{Z}} \mathbb{Q} = \mathbb{Q}$: every elliptic curve over $\mathbb{Q}$ has this property. You want to require an equality of L-series between the abelian variety and the modular form.