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 \(N\).