Rational Analytic Prismatization
This presentation introduces rational analytic prismatization, a geometric construction that unifies several central tools in p-adic geometry. By building overconvergent thickenings of perfectoid Fargues–Fontaine curves, the authors create a p-adic twistor stack whose perfect complexes encode relative varphi-Gamma modules and pro-étale cohomology. The talk explains how this construction geometrizes Sen theory and establishes a deep equivalence between seemingly distinct categories through locally analytic decompletion.Script
When you build a curve over a perfectoid space and then thicken it with overconvergent Frobenius data, you get more than just another geometric object. You get a space whose coherent sheaves encode the fundamental algebraic structures of p-adic Hodge theory.
The construction works through a pushout. Take the punctured Fargues-Fontaine disc over your perfectoid base, and attach copies of your nilperfectoid space indexed by the natural numbers, identifying Frobenius translates as you go. The disc accounts for the perfectoid geometry; the attached copies encode the overconvergent nilpotent directions.
Over the locus of zero divisors sits the Hodge-Tate stack, and it is not split. It forms a gerbe banded by the Breuil-Kisin-twisted overconvergent tangent bundle. This twist is intrinsic, cannot be removed without extra choices, and its presence controls the entire deformation theory near the Hodge-Tate divisor.
For smooth partially proper rigid spaces, the central theorem states that perfect complexes on the p-adic twistor stack are equivalent to perfect complexes on the relative Fargues-Fontaine curve. The two spaces differ geometrically, but the difference is invisible to dualizable objects. Pull back along the comparison map, and you get a fully faithful equivalence.
The proof mechanism is geometric Sen theory implemented through locally analytic decompletion. When a p-adic Lie group acts on a Berkovich space, passing from continuous to locally analytic quotients is prim and induces equivalence on perfect objects under the right hypotheses. This is not approximation. Locally analytic vectors are the correct coefficients for the twistor stack.
This construction places prismatization, pro-étale cohomology, relative varphi-Gamma modules, and geometric Sen theory into a single coherent framework. The integral theory over the p-adic integers remains open, but the rational picture is now clear. To learn more about this and other recent advances, visit EmergentMind.com and create your own lightning talks from the latest research.
