## 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.

This entry was posted in Mathematics and tagged , , , . Bookmark the permalink.