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On the Euler characteristic of $S$-arithmetic groups

Holger Kammeyer, Giada Serafini

TL;DR

The paper proves that for $S$-arithmetic subgroups of simply-connected simple algebraic groups with CSP, the sign of their Euler characteristic is determined by the $S$-congruence completion and is a profinite invariant, except possibly for type ${}^6 D_4$. The authors develop a framework using Poitou–Tate duality and Brauer–Witt invariants to relate local data across places, then perform a detailed Cartan-type analysis to show the parity $d( extbf{G})$ matches for corresponding groups, which fixes the sign when $ ext{χ}( ext{Γ}) eq0$. They further explain that CSP ensures the congruence and profinite completions are commensurable, enabling adelic rigidity arguments that yield an isomorphism of adèles and the same Cartan type, thus forcing equality of signs in many cases. The full argument culminates in a comprehensive classification by Cartan type, identifying precisely which types can yield zero, positive, or negative Euler characteristics, and highlighting the exceptional potential behavior in full triality ${}^6D_4$. The work extends prior profinite-invariant results to the $S$-arithmetic CSP context and clarifies when signs are robust under profinite equivalence, with implications for understanding higher $ ext{l}^2$-cohomology and the arithmetic geometry of lattices.

Abstract

We show that the sign of the Euler characteristic of an $S$-arithmetic subgroup of a simple algebraic group depends on the $S$-congruence completion only, except possibly in type ${}^6 D_4$. Consequently, the sign is a profinite invariant for such $S$-arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke--Raimbault--Sauer.

On the Euler characteristic of $S$-arithmetic groups

TL;DR

The paper proves that for -arithmetic subgroups of simply-connected simple algebraic groups with CSP, the sign of their Euler characteristic is determined by the -congruence completion and is a profinite invariant, except possibly for type . The authors develop a framework using Poitou–Tate duality and Brauer–Witt invariants to relate local data across places, then perform a detailed Cartan-type analysis to show the parity matches for corresponding groups, which fixes the sign when . They further explain that CSP ensures the congruence and profinite completions are commensurable, enabling adelic rigidity arguments that yield an isomorphism of adèles and the same Cartan type, thus forcing equality of signs in many cases. The full argument culminates in a comprehensive classification by Cartan type, identifying precisely which types can yield zero, positive, or negative Euler characteristics, and highlighting the exceptional potential behavior in full triality . The work extends prior profinite-invariant results to the -arithmetic CSP context and clarifies when signs are robust under profinite equivalence, with implications for understanding higher -cohomology and the arithmetic geometry of lattices.

Abstract

We show that the sign of the Euler characteristic of an -arithmetic subgroup of a simple algebraic group depends on the -congruence completion only, except possibly in type . Consequently, the sign is a profinite invariant for such -arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke--Raimbault--Sauer.
Paper Structure (6 sections, 14 theorems, 46 equations)

This paper contains 6 sections, 14 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Brauer--Witt invariants and Poitou--Tate duality
  4. Adelically isomorphic groups
  5. The sign of the Euler characteristic by Cartan type
  6. Full triality type

Key Result

Theorem 1

For $i=1,2$, let $k_i$ be number fields and let $S_i$ be finite sets of places of $k_i$ containing the infinite ones. Let $\Gamma_i \le \mathbf{G_i}$ be $S_i$-arithmetic subgroups of simply-connected simple non-triality $k_i$-groups with commensurable congruence completions. Then $\operatorname{sgn

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 7
  • Proposition 8
  • proof
  • Proposition 9
  • proof
  • proof : Proof of Theorem \ref{['thm:ses-a2nm1']}.
  • ...and 14 more