This drawing is an old drawing I made when I was preparing for my qualifying exam in my second year of grad school at Northwestern. It is the crystalline period map. The tower to the left is the “Lubin-Tate” tower, the deeper it goes the more level structure. In the upper right corner there is projective space, and the “cat like” creatures below are moduli stacks of curves. I always draw the moduli stack of elliptic curves as a cat because it has 2 stacky points (its ears) and a third stacky point if you compactify “the sock” of the fundamental domain (its tail). At the time I was working on what became my PhD thesis, of using moduli stacks of curves with level structure to understand actions of groups on the moduli stack of formal groups with level structure.

If I was braver then, I would have added this drawing to the paper which I wrote at an Arizona Winter School. I still think that paper is good resource for learning quickly about the Fargue Fontaine curve. A Global Crystalline Period Map

All of this is on my mind because I have had the great privledge and joy of returning to this topic after years away.

While we are sharing old things, in 2018 or 2019, Artem, Kolya and I made an outline of an approach to rational Chromatic Vanishing \( H^(J_h, W(k)) \times \mathbb{Q} \simeq H^(J_h, W(k)[[u_1, …, u_{h-1}]][u^{\pm}]) \times \mathbb{Q} \) using the two tower isomorphism which reduces the problem to directly calculating the $GL_h(\mathbb{Q}_p)$ action on constant functions on the Drinfeld projective space Chromatic Vanishing Approach.

At the time, the tools to compute the p-adic cohomology of the Drinfeld projective plane were not available to us so we ulitmately were scooped. I was initially sad about this because I felt then I could not prioritize thinking about these awesome and gorgeous concepts. However, recently I have begun with many others a spin-off project, which I am delighted and healed by. :)