I previously mentioned a problem concerning polynomials, whose motivation came from thinking about weight one forms and the inverse Galois problem for finite subgroups of I still like the polynomial problem, but I realized that I was confused about the intended application. Namely, given a weight one form with projective image there is certainly a unique minimal lift up to twist, but the images of the twists *also* automatically have image given by a central extension So, just by twisting, one can generate all such groups as Galois groups by starting with a minimal lift. More prosaically, every central extension of by a cyclic group is either a quotient of or of where is the Darstellungsgruppe of (which is So, to solve the inverse Galois problem for central extensions of , it suffices to solve it for That is not entirely trivial, but it is true.

I still think it’s an interesting problem to determine which extensions of by cyclic groups occur as the Galois groups of *minimally ramified up to twist* extensions, but that is not the same as the inverse Galois problem.