[Paper Review] On the middle convolution of local systems. With an Appendix by M. Dettweiler and S. Reiter

Michael Dettweiler|ArXiv.org|Oct 18, 2008

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper develops a motivic interpretation of middle convolution for lisse étale sheaves and perverse sheaves in étale cohomology, proving an independence-of-ℓ result and applying it to realize special linear groups as Galois groups over ℚ(t). In an appendix, it constructs a new motivic local system with monodromy dense in the exceptional group G₂, using iterative middle convolution and tensor operations on Kummer sheaves and character sheaves over ℤ[ζ₃, 1/3].

ABSTRACT

We study the middle convolution of local systems and realize special linear groups as Galois groups over the rationals. In the Appendix to this paper, written jointly with Stefan Reiter, we prove the existence of a new motivic local system with $G_2$-monodromy.

Motivation & Objective

• To provide a motivic interpretation of middle convolution in the étale cohomological setting, extending Katz’s theory to construct geometric Galois representations.
• To prove an independence-of-ℓ result for the determinant of middle convolution, enabling arithmetic applications.
• To construct new motivic local systems with prescribed monodromy groups, particularly dense in G₂, using convolution operations.
• To realize special linear groups as Galois groups over ℚ(t) via middle convolution of elementary sheaves.
• To establish the existence of a motivic local system with Zariski-dense monodromy in the exceptional group G₂, using rigidity and cohomological criteria.

Proposed method

• Define middle convolution of lisse étale sheaves via higher direct images with compact support and standard sheaf operations on A¹ × A¹.
• Use the intermediate extension j_!*, to define middle convolution as the image of the !-convolution map to the *-convolution map.
• Employ the Künneth formula and motivic interpretation of middle convolution (Theorem 2.6.1) to construct motivic local systems over ℤ[ζ₃, 1/3].
• Apply iterative middle convolution and tensor operations on Kummer sheaves and character sheaves associated to roots of unity.
• Use MAGMA computations to verify that monodromy matrices stabilize a 1-dimensional subspace in the third exterior power, implying G₂-monodromy.
• Apply the numerical criterion for physical rigidity (Katz, Thm. 1.1.2) and bireflection group classification (Springer, 1974) to confirm Zariski closure as G₂.

Experimental results

Research questions

• RQ1 Can the middle convolution of lisse étale sheaves be given a motivic interpretation in the context of ℓ-adic cohomology?
• RQ2 Does the determinant of the middle convolution satisfy independence-of-ℓ, and what does this imply for Galois representations?
• RQ3 Can middle convolution be used to construct motivic local systems with monodromy dense in exceptional algebraic groups like G₂?
• RQ4 Is it possible to realize SL(n) as a Galois group over ℚ(t) via middle convolution of elementary sheaves?
• RQ5 What conditions ensure that the monodromy of a middle convolution local system is Zariski-dense in a reductive group like G₂?

Key findings

• The middle convolution of lisse étale sheaves admits a motivic interpretation via the Künneth formula and Theorem 2.6.1, valid over ℤ[ζ₃, 1/3].
• An independence-of-ℓ result for the determinant of middle convolution is established, supporting the arithmetic nature of the construction.
• A new motivic local system on X(ℂ) is constructed whose monodromy representation has Zariski closure equal to G₂(ℚ̄_ℓ).
• The monodromy tuple of the constructed local system is shown to stabilize a 1-dimensional subspace in the third exterior power of ℚ̄_ℓ⁷, confirming G₂-structure.
• The local system is not physically rigid, as per Katz’s criterion, but is cohomologically non-rigid, indicating non-trivial deformation space.
• The construction realizes SL(n) as Galois groups over ℚ(t) for n ≥ 2, using iterative convolution of Kummer and character sheaves.