Number Fields Arising from Newforms - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:13:42Z http://mathoverflow.net/feeds/question/103545 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103545/number-fields-arising-from-newforms Number Fields Arising from Newforms Jeff H 2012-07-30T20:30:33Z 2012-07-30T22:48:24Z <p>It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$. </p> <p>In their 1995 <a href="http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf" rel="nofollow">paper</a> "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.</p> <p>This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:</p> <p>1) Can someone please provide a reference for this claim?</p> <p>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 $k=2$?</p> <p>Thank you!</p> <blockquote> <p>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!</p> </blockquote> http://mathoverflow.net/questions/103545/number-fields-arising-from-newforms/103551#103551 Answer by Stopple for Number Fields Arising from Newforms Stopple 2012-07-30T21:05:42Z 2012-07-30T21:32:00Z <p>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.</p> <p>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.</p> <p>This answer is not quite right because it doesn't explain how CM extensions can arise, but it's a start.</p> http://mathoverflow.net/questions/103545/number-fields-arising-from-newforms/103554#103554 Answer by Rob Harron for Number Fields Arising from Newforms Rob Harron 2012-07-30T21:34:28Z 2012-07-30T21:34:28Z <p>For (1), see Ribet's wonderful article <em>Galois representations attached to eigenforms with Nebentypus</em> (<a href="http://dx.doi.org/10.1007/BFb0063943" rel="nofollow">http://dx.doi.org/10.1007/BFb0063943</a>). It's proposition 3.2.</p>