Higher Hida theory for Hilbert modular varieties in the totally split case

Abstract

We study p-adic properties of the coherent cohomology of some automorphic sheaves on the Hilbert modular variety X for a totally real field F in the case where the prime p is totally split in F . More precisely, we develop higher Hida theory à la Pilloni, constructing, for 0 ≤ q ≤ [F : Q], some modules M q which p-adically interpolate the ordinary part of the cohomology groups H q (X, ω κ ), varying the weight κ of the automorphic sheaf. Contents 1. Introduction 1 2. Preliminaries 4 2.1. Hilbert modular varieties and moduli interpretation 4 2.2. Compactifications 6 2.3. Automorphic vector bundles 6 3. Hecke operators and Hasse invariants 10 3.1. Hecke operators 10 3.2. Partial Hasse invariants and the Goren-Oort stratification 12 4. Higher Hida theory 13 4.1. Mod p theory 13 4.2. Characteristic zero theory 17 4.3. Duality 27 References 29

On the Bloch--Kato conjecture for GSp(4) x GL(2)

“…Even in the ordinary case, to prove the Bloch-Kato conjecture for all of the critical twists, we would need a version of higher Hida (rather than Coleman) theory for GSp 4 with both r 1 and r 2 varying. Such a theory is not available at present, although analogous results for Hilbert modular groups have been announced by Giada Grossi [Gro21].…”