[Paper Review] Rigid local systems and relative motives with Galois group G_2

Michael Dettweiler, Stefan Reiter|May 15, 2006

Algebraic Geometry and Number Theory8 references18 citations

TL;DR

This paper constructs rigid local systems with monodromy dense in the exceptional algebraic group $G_2$ using N. Katz's middle convolution functor. It proves the existence of motives with motivic Galois group of type $G_2$, providing a partial solution to a question of Serre and supporting Grothendieck's standard conjectures via étale cohomology and Tannaka duality.

ABSTRACT

We classify rigid local systems of rank 7 whose monodromy group is dense in the simple algebraic group of type G2. This leads to motives with Galois group G2.

Motivation & Objective

• To establish the existence of rigid local systems on $\mathbb{P}^1 \setminus \{0,1,\infty\}$ with monodromy group Zariski dense in $G_2$.
• To demonstrate that such rigid local systems arise from the middle convolution functor $\mathrm{MC}_\chi$ applied to rank-one sheaves.
• To derive the existence of motives with motivic Galois group of type $G_2$ under Grothendieck's standard conjectures.
• To provide a partial answer to Serre's question on the existence of motives with exceptional motivic Galois groups.

Proposed method

• Apply the middle convolution functor $\mathrm{MC}_\chi$ to irreducible étale rigid local systems to reduce them to rank-one sheaves.
• Use the Katz existence algorithm to test whether a given set of local monodromy conditions lifts to an irreducible rigid local system.
• Construct a rank-7 étale rigid local system $\mathcal{H}(\varphi,\eta)$ on $\mathbb{P}^1_k \setminus \{0,1,\infty\}$ with specified local monodromy types at $0$, $1$, and $\infty$.
• Use the comparison isomorphism between étale and singular cohomology to relate the monodromy representation to the topological fundamental group.
• Construct a family of motives $M_s = N_s(3)$ from a family of varieties $X/S$ via the Künneth projector and relative cohomology.
• Apply Tannaka duality to show that the motivic Galois group of $M_s$ is isomorphic to $G_2$ for all $s \in \mathbb{Q} \setminus \mathrm{Exc}$.

Experimental results

Research questions

• RQ1 Does there exist a rigid local system on $\mathbb{P}^1 \setminus \{0,1,\infty\}$ whose monodromy group is Zariski dense in $G_2$?
• RQ2 Can such a rigid local system be constructed using the middle convolution functor $\mathrm{MC}_\chi$?
• RQ3 Do motives exist whose motivic Galois group is of type $G_2$?
• RQ4 Is the existence of such motives consistent with Grothendieck's standard conjectures?
• RQ5 Can the motivic Galois group of a motive be realized as $G_2$ via geometric constructions in algebraic geometry?

Key findings

• The paper constructs an étale rigid local system $\mathcal{H}(\varphi,\eta)$ of rank 7 on $\mathbb{P}^1_k \setminus \{0,1,\infty\}$ with monodromy group Zariski dense in $G_2(\bar{\mathbb{Q}}_\ell)$ under specified conditions on $\varphi$ and $\eta$.
• The local monodromy at $0$ is $-\mathbf{1} \oplus -\mathbf{1} \oplus -\mathbf{1} \oplus -\mathbf{1} \oplus \mathbf{1} \oplus \mathbf{1} \oplus \mathbf{1}$, at $1$ is $\mathrm{U}(2) \oplus \mathrm{U}(2) \oplus \mathrm{U}(3)$, and at $\infty$ varies depending on $\varphi$ and $\eta$, with cases including $\mathrm{U}(7)$ and direct sums of unipotent representations.
• The monodromy representation of the local system $\mathcal{H}(\mathbf{1},\mathbf{1})$ is Zariski dense in $G_2(\bar{\mathbb{Q}}_\ell)$, as shown via the comparison isomorphism and the Tannaka reconstruction of the motivic Galois group.
• For all $s \in \mathbb{Q} \setminus \mathrm{Exc}$, the motivic Galois group of the motive $M_s = N_s(3)$ is isomorphic to $G_2$, with $N_s$ arising from the $\ell$-adic cohomology of a family of varieties.
• The motive $A_s$ dual to the character part of $N_s$ is shown to be the Tate twist $({\rm Spec\,}(s), {\rm Id}, 3)$, confirming that $M_s$ has motivic Galois group $G_2$.
• Under Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type $G_2$ is established, providing a partial answer to Serre’s question on exceptional motivic Galois groups.