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: using moduli stacks of curves with marked points to understand moduli stacks 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 \(H^*(J_h, W(k)) \times \mathbb{Q} \simeq H^*(J_h, W(k)[[u_1, ..., u_{h-1}]][u^{\pm}]) \otimes \mathbb{Q}\) 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. :)