Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:33:31Z http://mathoverflow.net/feeds/question/101274 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101274/are-coefficients-of-maass-forms-of-eigenvalue-1-4-known-to-be-algebraic Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic? unknown (google) 2012-07-04T00:52:03Z 2012-07-04T05:24:33Z <p>I would really like to know whether the following famous conjecture has been solved. I've read in a few places that it has been solved, but I have been unable to find a reference. I do know that there was once a very credible proof which was believed for a while and then turned out to be false. The statement is:</p> <blockquote> <p>If $f:\mathbb H/\Gamma(N)\to\mathbb R$ is of moderate growth in the cusps and satisfies $\Delta f+\frac 14f=0$, then the Hecke eigenvalues $\lambda_p(f)$ are algebraic.</p> </blockquote> <p>There is of course a much stronger conjecture (still unsolved) that there is a Galois representation $\rho:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\to GL(2,\mathbb C)$ such that $\lambda_p(f)=\operatorname{tr}\rho(\operatorname{Fr}_p)$. This would imply the weaker conjecture above.</p> <p>It was once believed that this stronger conjecture was solved by <a href="http://www.ams.org/mathscinet-getitem?mr=1012167" rel="nofollow">Blasius and Ramakrishnan</a> (and see a <a href="http://www.ams.org/mathscinet-getitem?mr=920047" rel="nofollow">second paper</a>), but their proof relied on a statement (which they could not prove, but which was widely believed to be true) which turned out to be false. I thought at some point, though, that, perhaps because of work of Richard Taylor, that the weaker boxed statement above had been proven later using different methods. I have been unable to locate a reference, though, and many papers actually refer to the claimed proof of B&amp;R.</p> <p>Is the boxed statement known to be true (and does someone know a good reference for the proof)?</p> http://mathoverflow.net/questions/101274/are-coefficients-of-maass-forms-of-eigenvalue-1-4-known-to-be-algebraic/101285#101285 Answer by David Hansen for Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic? David Hansen 2012-07-04T05:24:33Z 2012-07-04T05:24:33Z <p>Your boxed statement is an open problem. Blasius and Ramakrishnan did not rely on a widely believed statement which turned out to be false. Their argument accidentally conflated two L-packets for $GSp_4(\mathbb{R})$ which are in fact distinct, due to a miscalculation of the central character of one of the L-packets in question. This mistake was discovered several years later by Henniart, I believe - see MR1157812.</p>