The image of J and p-adic geometry

For a prime p, Bhatt, Lurie, and Drinfeld constructed the “prismatization” of a p-adic formal scheme; this is a stack which computes prismatic cohomology, which is a “universal” cohomology theory for p-adic formal schemes. I will describe joint work with Hahn, Raksit, and Yuan (building on work of Hahn-Raksit-Wilson), in which we give a new construction of prismatization using the methods of homotopy theory (in particular, the theory of topological Hochschild homology, aka THH). The case when R is Z_{p} turns out to be particularly interesting, and I will discuss joint work with Raksit which describes a construction of THH(Z_{p}) for odd primes p in terms of a very classical object in homotopy theory called the “image-of-J
spectrum” studied by Adams. This plays the same role for prismatic cohomology as the usual commutative ring Z_{p} plays for crystalline cohomology. It gives an alternative perspective on results of Bhatt and Lurie, and is also related to Lurie’s “prismatization of F_{1}”.