I often ask mathematical questions on this blog that I don’t know how to answer. Sometimes my smart readers are able to answer the questions I ask. Surely they deserve some recognition for this? Here are two such occasions which come to mind (one very recent):
In this post, I asked whether there are infinitely many integers n such that all the odd divisors of (n^2 + 1) *not* of the form 1 \bmod 2^m for large enough fixed m, and asked whether that was an open problem. The answer: it was then, but no longer! It has now been answered by Soundararajan and Brüdern in Theorem 4 of this preprint. Problem solved!
In this post, I was looking at tables of George Schaeffer at non-liftable weight 1 modular forms of level \Gamma_1(N) for various quadratic characters, and noting that often there were forms with large prime factors. I said:
I said “However, something peculiar happens in the range of the tables, namely, there is not a single example with N prime. This leads to the (incredibly) vague question: can this be predicted in advance?”
But later GB pointed out to me that when N is not prime, and the corresponding quadratic character (in the tables) is not divisible by a prime q, then the Galois representation at the auxiliary prime q need not be unramified (it can be of Steinberg type) and the corresponding Galois representations can have have significantly larger root discriminant — the ramification index at those primes is e_q = \ell for the residue characteristic \ell rather than e_q = 2. And indeed, looking more closely at the tables, most of the big primes \ell for which there exist non-liftable forms of level (N,\chi) occur when the conductor of \chi strictly smaller than N.