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It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ordinary integer. Of course, there are other induced splittings by the Fricke involution, however the finest one is by the Hecke eingenfunctions. However I´ve heard that each $E(f)$ has a real multiplication. Is it true? Any references?

Thanks in advance.

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This is true. More precisely, any $E(f)$ has real multiplication by the field $K_f$ generated by the coefficients of $f$, and this field is a totally real number field of degree over $\mathbb Q$ equal to the dimension of $E(f)$. This is due to Shimura and proven in his book "Introduction to the arithmetic theory of automorphic functions" (see chapter 7). The arguments is also explained in more modern references, such as the book on modular forms by Diamond and Shurman and the text on Fermat's last theorem by Diamond, Darmon and Taylor.

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  • $\begingroup$ Thanks. By the way, since you work on this field, do you know a reference that explains Deligne's "Travaux de Shimura"? $\endgroup$
    – user40276
    Commented Aug 2, 2014 at 2:50
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    $\begingroup$ @user40276: How about Milne's text on Shimura varieties? jmilne.org/math/xnotes/index.html $\endgroup$
    – Vesselin Dimitrov
    Commented Aug 2, 2014 at 8:40
  • $\begingroup$ @VesselinDimitrov Thanks, I already knew this reference. Actually I was searching for a reference that I've heard long time ago that has almost 500 pages and focus on explaining everything in Deligne's paper. $\endgroup$
    – user40276
    Commented Aug 15, 2014 at 8:10

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