# Modular forms reference

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.

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MAy I ask, how did you find the book of shimura? Especially for a beginner in automorphic functions.. I have the book, I have read a chapter of it, but then stopped as this is not really my research area. But somehow, I get the impression that the approach of the book is not a standard one.. at least not for a beginners (though for me, it was still quite readable) –  Jose Capco Oct 31 '09 at 21:08
It was the reference that Ribet gave in his paper 'Endomorphism algebras of abelian varieties attached to newforms of weight 2'. –  Ben Linowitz Oct 31 '09 at 21:29
Shimura is the standard reference. Also he is the prover of all this stuff and creator of many streams of thought in the subject. Unfortunately his book is unreadable. See my answer below and the comments along with it. –  Anweshi Jan 4 '10 at 14:49

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.

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Just to add one more thing to what Pete said: the variety A_f that one normally attaches to f might have endomorphism ring bigger than an order in the coefficient field of f: for example if E is an elliptic curve with complex multiplication, the associated modular form still only has coefficients in Z. The correct statement is that the endomorphism ring contains an order in the coefficient field

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Right, but I said End_Q(E) -- by which I meant (but didn't explain: sorry!) the subring of endomorphisms defined over Q. This is indeed always equal to an order in the number field generated by the Fourier coefficients. The full endomorphism ring may well be larger, as happens in the elliptic curve case if(f) there is complex multiplication. –  Pete L. Clark Nov 4 '09 at 15:10
Yes Pete---apologies, you're dead right. –  Kevin Buzzard Nov 9 '09 at 20:24