Griffiths-Schmid conditions for automorphic forms via characteristic $p$

TL;DR

This paper proves vanishing results for automorphic forms on Hodge-type Shimura varieties in both characteristic 0 and p, using algebraic methods, and verifies a conjecture for certain unitary PEL Shimura varieties.

Contribution

It establishes Griffiths-Schmid conditions for automorphic forms in characteristic 0 via algebraic methods and proposes a conjecture for characteristic p, verified in specific cases.

Findings

Vanishing results for automorphic forms in characteristic 0 and p.

Proof that weights satisfy Griffiths-Schmid conditions in characteristic 0.

Verification of the conjecture for unitary PEL Shimura varieties of signature (n-1,1).

Abstract

We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.