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It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.

In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K_f$. They do claim, however, that $K_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.

This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:

1) Can someone please provide a reference for this claim?

2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K_f$ for some $f$? What if we restrict our attention to weight $k=2$?

Thank you!

Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out! Also, I originally claimed the paper was published in 2007, but Kevin Buzzard pointed out it was published in 1995. Thanks Kevin!

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2 Answers

up vote 7 down vote accepted

For (1), see Ribet's wonderful article Galois representations attached to eigenforms with Nebentypus (http://dx.doi.org/10.1007/BFb0063943). It's proposition 3.2.

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@Rob: Thank you! This is perfect. –  Jeff H Jul 30 '12 at 22:28
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See also item 53 in math.berkeley.edu/~ribet/Articles. –  Chandan Singh Dalawat Jul 31 '12 at 3:29
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It's not true for any old modular form. Since the forms live in a vector space over $\mathbb C$, you can achieve any complex number as a coefficient.

Here's a partial answer to what is true. You need to have a cusp form that is an eigenfunction of the Hecke operators, normalized so the leading coefficient is $1$. Since the Hecke operators are self adjoint in the Peterson (sp?) inner product, the eigenvalues are real, and one can show these are the coefficients in the $q$ expansion as follows: for $p$ prime, the $m$th coefficient of $T_p f$ is $a_{mp}$, for all $m$, more or less from the definition of $T_p$. This is also $\lambda_p a_m$, and from this and $a_1=1$ one deduces $a_p=\lambda_p$ (take $m=1$.) The general case follows from the recursion for powers of primes, and multiplicativity.

This answer is not quite right because it doesn't explain how CM extensions can arise, but it's a start.

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Hi Stopple, good point. It's a bad habit, but in my mind, when I say modular form I (almost) always mean "normalized eigenform", or even just "newform", but certainly the way I stated my question is incorrect. I'll fix my question to make it more precise. Thanks for pointing this out! –  Jeff H Jul 30 '12 at 21:15
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The Hecke operators are self-adjoint for $\Gamma_0(N)$; however when dealing with forms with a character (e.g. eigenforms) the Hecke operators are only self-adjoint "up to a twist", from which one deduces that $\overline{a}_n=\chi(n)^{-1}a_n$, and the claim follows from this. PS Jeff H: I remember reading that paper in 1995, which makes me a bit skeptical about the claim that it was published in 2007! –  Kevin Buzzard Jul 30 '12 at 21:38
    
@Kevin Buzzard: Oh! You're absolutely right -- I see on Darmon's website that it was indeed published in 1995. I guess the tex file was recompiled in 2007, leading to the date I cited from this version of the paper: www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf Thanks! –  Jeff H Jul 30 '12 at 22:25
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