[Paper Review] On maximal tori of algebraic groups of type G_2

Constantin N. Beli, Philippe Gille+1 more|arXiv (Cornell University)|Nov 25, 2014

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper studies maximal tori in algebraic groups of type $G_2$ via their connection to octonion algebras over a field $k$. It provides a cohomological criterion for the existence of $k$-embeddings of a rank-two torus into $\mathrm{Aut}(C)$, using Galois cohomology and the Weyl group $W_0 \cong \mathbb{Z}/2\mathbb{Z} \times S_3$, with a complete solution when the associated cubic extension is not a field and a complex criterion when it is. The work also constructs counterexamples to Totaro’s question on rational points in homogeneous spaces over Laurent series fields.

ABSTRACT

Given an octonion algebra over a field k, its automorphism group G is an algebraic semisimple k-group of type G_2. We study the maximal tori of G in terms of the algebra C.

Motivation & Objective

• To classify maximal tori of algebraic groups of type $G_2$ over a field $k$ using the structure of octonion algebras.
• To determine when a given $W_0$-torsor, corresponding to a pair $(k', l)$ of étale algebras, arises as the type of a $k$-embedding of a torus into $\mathrm{Aut}(C)$ for an octonion algebra $C$.
• To provide a cohomological criterion for the existence of $k$-rational points on certain homogeneous spaces under $G_2$, particularly in the context of Totaro’s question on zero-cycles and rational points.
• To establish that for number fields, the Hasse principle holds for these varieties, making the classification effective.

Proposed method

• Use the fact that $G_2$ arises as the automorphism group of an octonion algebra $C$ over $k$, and study maximal tori via their action on the root system of $G_{k_s}$.
• Associate to each $k$-embedding $i: T \to G$ a cohomology class $\mathrm{type}(T,i) \in H^1(k, W_0)$, where $W_0 \cong \mathbb{Z}/2\mathbb{Z} \times S_3$ is the Weyl group of $G_2$.
• Relate the existence of such embeddings to the existence of $k$-points on a homogeneous space $X$ under $G$, constructed from a pair $(k', l)$ of étale algebras.
• For the case where $l$ is not a field, use the structure of subgroups of type $A_2$ in $G$ to reduce the problem to known results on unitary groups and apply the theory of Knus, Haile, Rost, and Tignol.
• For the case where $l$ is a field, derive a complex criterion (Theorem 5.2.6) based on the geometry of $A_2$-subgroups and Galois cohomology.
• Construct counterexamples to Totaro’s question in the case $d=1$ over Laurent series fields $k((t))$, using a filtration argument on $T(k'[[t]])$ and Hilbert’s Theorem 90 to show the existence of quadratic and cubic points without $k$-rational points.

Experimental results

Research questions

• RQ1 Given an octonion algebra $C$ over a field $k$, when does a given $W_0$-torsor $[(k', l)]$ arise as the type of a $k$-embedding of a maximal torus into $\mathrm{Aut}(C)$?
• RQ2 What conditions on the étale algebras $k'$ and $l$ ensure the existence of a $k$-rational point on the associated homogeneous space $X$ under $G_2$?
• RQ3 Can one construct, for a field $k$ of the form $k((t))$, a homogeneous space $X$ under $G_2$ with a zero-cycle of degree 1 but no $k$-rational point?
• RQ4 Does the Hasse principle hold for the varieties parametrizing maximal tori of $G_2$ over number fields?
• RQ5 How does Bruhat-Tits theory relate to the rational conjugacy of tori in $G_2$-groups over local fields?

Key findings

• For a non-field cubic extension $l$, the paper gives a complete and explicit criterion for the existence of a $k$-embedding of a torus of type $(k', l)$ into $\mathrm{Aut}(C)$, based on the structure of $A_2$-subgroups and Galois cohomology.
• When $l$ is a field, the paper provides a complex criterion (Theorem 5.2.6) involving the geometry of $A_2$-subgroups and the cohomology of unitary groups, which determines whether a given $W_0$-torsor lifts to a $k$-embedding.
• Over Laurent series fields $k((t))$, the paper constructs a homogeneous space $X$ under $G_2$ that has a quadratic point and a cubic point but no $k$-rational point, thus providing new counterexamples to Totaro’s question in the case $d=1$.
• For number fields, the Hasse principle holds for the varieties parametrizing maximal tori of $G_2$, and the paper gives an effective method to compute the types of all maximal tori, as illustrated in Example 6.4 for the compact form of $G_2$ over $\mathbb{Q}$.
• The paper establishes a link between Bruhat-Tits theory and Galois cohomology by showing that a $G^{ ext{ad}}_K$-torsor $E$ has a $K_{nr}$-point if and only if it arises from a $k$-torus embedding into $G^\prime$, generalizing Steinberg’s theorem to the local field setting.
• The specialization map $H^1(\Gamma, T(k'[[t]])) \to H^1(\Gamma, T(k'))$ is an isomorphism, proven via a filtration argument on the units of the power series ring, which allows reduction to Hilbert’s Theorem 90 and establishes the key exact sequence in the proof of the main theorem.