Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties (Q411725)

Consider a compact PEL-type Shimura variety \(M\) at a neat level, a weight \(\mu\) and a prime number \(p\). Suppose that the linear-algebraic data defining \(M\) are unramified at \(p\), the level is maximal hyperspecial at \(p\), and that (i) \(\mu\) is sufficiently regular, (ii) \(p\) is larger then a certain bound depending solely on \(\mu\) and the prime-to-\(p\) level. The authors prove that the Betti cohomology of \(M\) with coefficients in the \(\mathbb{Z}_p\)-module associated to \(\mu\) is concentrated in the middle degree; moreover, it has no \(p\)-torsion. The main ingredients of the proof include the theory of automorphic sheaves on \(M\), together with a vanishing theorem of Deligne-Illusie. The general non-compact case is deferred to [\textit{K.-W. Lan} and \textit{J. Suh}, Adv. Math. 242, 228--286 (2013; Zbl 1276.11103)], which is much more involved.NEWLINENEWLINEAs explained in this paper, the conditions on \(\mu\) and \(p\) alluded to above can be effectively verified in practice. Another highlight is that the vanishing theorem does not require any assumption on the associated Galois representations, which is often subtle and difficult to check.