Origin of Research Interests

My interest in F-isocrystals and l-adic local systems arose from my Ph.D. thesis. Motivated by results of Mochizuki, I tried to characterize Shimura curves over a finite field using purely group-theoretic data: the notion of an etale correspondence without a core. These exhibit many formal similarities with Hecke correspondences of Shimura curves. For instance, from a correspondence without a core one can construct an infinite graph with a large group of algebraic automorphisms; in the case of a Hecke correspondence of Shimura curves, this specializes to the action of PSL₂(Q_p) on its building.

A more elaborate group-theoretic hypothesis led to the following question. Let X ← Z → X be an etale correspondence without a core, and suppose there is an SL₂(Q_l) local system on X such that the two pullbacks to Z are isomorphic as local systems. Then is the whole package related to a Hecke correspondence of Shimura curves? The example of modular curves with Igusa level structures shows that "related to" is essential: the correspondence may not simply deform to characteristic zero.

Work of Tomoyuki Abe completes Deligne's companions conjecture for curves by proving a p-adic Langlands correspondence over a finite field. Using Abe's results together with foundational work of de Jong, we translated the condition on local systems to a condition on associated p-divisible groups. Under sufficiently auspicious circumstances, the correspondence together with the p-divisible groups deforms to characteristic zero, and Mochizuki's theorem implies that the geometry is at least related to a Hecke correspondence of Shimura curves.

The work of Lafforgue and Abe shows that the local systems and overconvergent F-isocrystals that occur here are motivic; this was one of the original motivations for the companions conjecture. Later work of Deligne, Drinfeld, Abe-Esnault, and Kedlaya proves much of the higher-dimensional companions picture. Since finishing my Ph.D., I have mostly thought about this geometricity problem and the conjectures it suggests. I view this as analogous to Simpson's conjecture that rigid local systems on smooth complex algebraic varieties are motivic.