if they IPO can Nvidia buy a ton of shares and indirectly buy Arm?
I'm still unclear why Nvidia would want to control Arm. If they wanted an architecture they could extend as they wished, they can use RISC-V (which they are already doing in a small way).
I think nvidia is worried about being shut out of the x86 space so there's nowhere to sell their products. So they had a long term play to guide ARM development as an alternative platform. I could see that platform being useful as a hedge, but also giving nvidia a better way of competing in consoles and if they were really successful maybe eventually turning Intel/AMDs x86 advantage into a weakness.
Back when Intel killed off all third party motherboard chipsets nvidia won a lawsuit against Intel where they forced Intel to ensure a pcie bus for GPUs on all their platforms. That provision expired a few years ago. If nvidia is smart (and I believe they are) they've never let go of that worry that APUs might shove them out of the market. Their best bet is to remain so much better than the competition they cannot be ignored, but it would be wise to have a backup plan.
Nvidia have some really good high performance Arm cores that they've developed and sold (Denver, Carmel). Continuing to invest in that area makes some sense for Nvidia today, but it would make a lot of sense if the products they're paying to develop were official Arm products sold through the Arm IP sales channel. In other words, it would increase Nvidia's return on investment on work they're already doing.
(on the other hand, the amount they would have ended up paying for Arm might be fully justified by the value of the company but it was really, really high)