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.

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