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Weights of mod $p$ automorphic forms and partial Hasse invariants

Wushi Goldring, Naoki Imai, Jean-Stefan Koskivirta

TL;DR

This work introduces the zip cone $C_{ ext{zip}}$ attached to a cocharacter datum $(G,oldsymbol{mu})$ and studies its relation to cones arising from automorphic forms in characteristic zero and characteristic $p$. It proves the general inclusion ${f C}(C) o C_{ ext{zip}}$, providing evidence for the conjectured equality ${f C}(ar{F}_p)=C_{ ext{zip}}$ in good-reduction Hodge-type Shimura varieties. A central achievement is a precise criterion for when the zip cone is generated by partial Hasse invariants, characterizing those $(G,oldsymbol{mu})$ of Hasse-type via equivalences involving Griffiths–Schmid cones and Frobenius actions. The paper also constructs natural mod $p$ automorphic forms from highest-weight vectors, and develops a framework to transfer cone questions to split groups via Weil restriction, with detailed examples for unitary and orthogonal groups. Collectively, these results advance understanding of weight cones across characteristic $0$ and $p$, and illuminate when Hasse invariants suffice to capture all automorphic weights.

Abstract

For a connected, reductive group $G$ over a finite field endowed with a cocharacter $μ$, we define the zip cone of $(G,μ)$ as the cone of all possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This cone is conjectured to coincide with the cone of weights of characteristic $p$ automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic $0$ automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.

Weights of mod $p$ automorphic forms and partial Hasse invariants

TL;DR

This work introduces the zip cone attached to a cocharacter datum and studies its relation to cones arising from automorphic forms in characteristic zero and characteristic . It proves the general inclusion , providing evidence for the conjectured equality in good-reduction Hodge-type Shimura varieties. A central achievement is a precise criterion for when the zip cone is generated by partial Hasse invariants, characterizing those of Hasse-type via equivalences involving Griffiths–Schmid cones and Frobenius actions. The paper also constructs natural mod automorphic forms from highest-weight vectors, and develops a framework to transfer cone questions to split groups via Weil restriction, with detailed examples for unitary and orthogonal groups. Collectively, these results advance understanding of weight cones across characteristic and , and illuminate when Hasse invariants suffice to capture all automorphic weights.

Abstract

For a connected, reductive group over a finite field endowed with a cocharacter , we define the zip cone of as the cone of all possible weights of mod automorphic forms on the stack of -zips. This cone is conjectured to coincide with the cone of weights of characteristic automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.
Paper Structure (33 sections, 50 theorems, 166 equations, 2 tables)

This paper contains 33 sections, 50 theorems, 166 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Definitions
  4. Cocharacter datum
  5. Frames
  6. Representation theory
  7. Vector bundles on quotient stacks
  8. Definition
  9. Partial Hasse invariants
  10. The zip cone
  11. Example: Hilbert--Blumenthal Shimura varieties
  12. General setting
  13. Previous results
  14. Norm of the highest weight vector
  15. Partial Hasse invariant cone, Griffiths--Schmid cone
  16. ...and 18 more sections

Key Result

Lemma 2.2.3

Let $\mu \colon \mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter, and let ${\mathcal{Z}}_\mu$ be the attached zip datum. Assume that $(B,T)$ is a Borel pair defined over $\mathbb{F}_q$ such that $B\subset P$. Define the element Then $(B,T,z)$ is a frame for ${\mathcal{Z}}_\mu$.

Theorems & Definitions (101)

  • Remark 2.2.1
  • Remark 2.2.2
  • Lemma 2.2.3: Goldring-Koskivirta-zip-flags
  • Theorem 2.2.4: Pink-Wedhorn-Ziegler-zip-data
  • Lemma 2.4.1: Koskivirta-Wedhorn-Hasse
  • Lemma 2.4.2: Imai-Koskivirta-vector-bundles
  • Theorem 2.4.3: Imai-Koskivirta-vector-bundles
  • Corollary 2.4.4
  • Proposition 2.5.1: Imai-Koskivirta-partial-Hasse
  • Definition 2.5.2
  • ...and 91 more