Control theorems for Hilbert modular varieties
Canadian mathematical bulletin, Tome 68 (2025) no. 2, pp. 530-549
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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 (2023, Spherical varieties and p-adic families of cohomology classes); another important ingredient in our proof is the recent work of Caraiani–Tamiozzo (2023, Compositio Mathematica 159, 2279–2325) 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.
Mots-clés :
Control theorems, Hilbert modular varieties, Hida theory
Sheth, Arshay. Control theorems for Hilbert modular varieties. Canadian mathematical bulletin, Tome 68 (2025) no. 2, pp. 530-549. doi: 10.4153/S0008439524000791
@article{10_4153_S0008439524000791,
author = {Sheth, Arshay},
title = {Control theorems for {Hilbert} modular varieties},
journal = {Canadian mathematical bulletin},
pages = {530--549},
year = {2025},
volume = {68},
number = {2},
doi = {10.4153/S0008439524000791},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000791/}
}
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