Question

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:

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.

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.

It was once believed that this stronger conjecture was solved by Blasius and Ramakrishnan (and see a second paper), 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&R.

Is the boxed statement known to be true (and does someone know a good reference for the proof)?

Answer 1

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.