## Central Extensions, Updated

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 $\mathrm{GL}_2(\mathbf{C}).$ 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 $A_5,$ 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 $A_5.$ 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 $A_5$ by a cyclic group is either a quotient of $A_5 \times \mathbf{Z}$ or of $\widetilde{A}_5 \times \mathbf{Z}$ where $\widetilde{A}_5$ is the Darstellungsgruppe of $A_5$ (which is $\mathrm{SL}_2(\mathbf{F}_5)).$ So, to solve the inverse Galois problem for central extensions of $A_5$, it suffices to solve it for $\mathrm{SL}_2(\mathbf{F}_5).$ That is not entirely trivial, but it is true.

I still think it’s an interesting problem to determine which extensions of $A_5$ 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.