Control Theorems for Hilbert Modular Varieties
arXiv:2403.04738 · doi:10.4153/S0008439524000791
Abstract
We prove an exact control theorem, in the sense of Hida theory, for the ordinary part of the middle degree étale cohomology of certain Hilbert modular varieties, after localizing at a suitable maximal ideal of the Hecke algebra. Our method of proof builds upon the techniques introduced by Loeffler-Rockwood-Zerbes; another important ingredient in our proof is the recent work of Caraiani-Tamiozzo on the vanishing of the étale cohomology of Hilbert modular varieties with torsion coefficients outside the middle degree. This work will be used in forthcoming work of the author to show that the Asai-Flach Euler system corresponding to a quadratic Hilbert modular form varies in Hida families.
16 pages; updated version incorporating suggestions from the referee. In particular, we have added a vanishing result for cohomology with non-trivial coefficients (Corollary 4.8) as an application of our results