[Paper Review] On the middle convolution

Michael Dettweiler, Stefan Reiter|ArXiv.org|May 22, 2003

Algebraic Geometry and Number Theory11 references12 citations

TL;DR

This paper provides a cohomological interpretation of the algebraic middle convolution functor $MC_{ heta}$ on local systems over the punctured complex line, establishing a Riemann-Hilbert correspondence via higher direct image sheaves. It proves that $MC_{ heta}(\mathcal{F}) \cong R^1\bar{p}_2_* (j_* (\mathrm{pr}_1^*\mathcal{F} \otimes q^*\mathcal{L}_\theta))$, enabling an algorithmic construction of Fuchsian systems for rigid local systems and verifying the Grothendieck-Katz $p$-curvature conjecture for new classes of differential equations.

ABSTRACT

We present a cohomological interpretation of the middle convolution functor MC and find an explicit Riemann-Hilbert correspondence for MC_λ. This leads to an algorithm for the construction of Fuchsian systems which correspond to irreducible rigid local systems under the Riemann-Hilbert correspondence. Also, the effect of MC_λon the p-curvatures ist determined and new examples of differential systems are found which satisfy the Grothendieck-Katz p-curvature conjecture.

Motivation & Objective

• To provide a cohomological interpretation of the algebraic middle convolution functor $MC_{\lambda}$, mirroring Katz’s $l$-adic middle convolution.
• To establish a Riemann-Hilbert correspondence for $MC_{\lambda}$, translating it into a differential system with regular singularities.
• To develop an algorithm for constructing Fuchsian systems corresponding to irreducible rigid local systems.
• To analyze the behavior of $MC_{\lambda}$ on $p$-curvatures and verify the Grothendieck-Katz $p$-curvature conjecture for new examples.

Proposed method

• Use of singular cohomology and sheaf-theoretic constructions on the fibration $p_2: E \to X$, where $E = \{(x,y) \in \mathbb{C}^2 \mid x,y \ne t_i, x \ne y\}$.
• Application of the higher direct image sheaf $R^1\bar{p}_2_* (j_* (\mathrm{pr}_1^*\mathcal{F} \otimes q^*\mathcal{L}_\lambda))$ to realize $MC_\lambda(\mathcal{F})$.
• Employment of Pochhammer contours and twisted bases to model monodromy and convolution via crossed homomorphisms.
• Construction of differential systems in Okubo normal form to analyze the effect of $MC_\lambda$ on the system's structure.
• Use of gauge transformations and block matrix decompositions to simplify the system and analyze nilpotency.
• Application of the Leray spectral sequence and motivic Galois group theory to verify the $p$-curvature conjecture for certain systems.

Experimental results

Research questions

• RQ1 How can the algebraic middle convolution $MC_\lambda$ be interpreted cohomologically in the complex setting?
• RQ2 What is the explicit Riemann-Hilbert correspondence for $MC_\lambda$ on local systems over $\mathbb{C} \setminus \{t_1,\dots,t_r\}$?
• RQ3 Can $MC_\lambda$ be used to algorithmically construct Fuchsian systems for irreducible rigid local systems?
• RQ4 What is the behavior of $MC_\lambda$ on $p$-curvatures, and does it preserve the Grothendieck-Katz $p$-curvature conjecture?
• RQ5 Under what conditions is the convolution of a globally nilpotent system also globally nilpotent?

Key findings

• The middle convolution $MC_\lambda(\mathcal{F})$ is isomorphic to the higher direct image sheaf $R^1\bar{p}_2_* (j_* (\mathrm{pr}_1^*\mathcal{F} \otimes q^*\mathcal{L}_\lambda))$, providing a cohomological realization of the functor.
• The construction yields an algorithm to produce Fuchsian systems with regular singularities whose solution sheaf is $MC_\lambda(\mathcal{F})$ for any given rigid local system $\mathcal{F}$.
• For $\mu = -1$, the convolution $c_{-1}(a(\mathfrak{p}))^{k+1} = 0$ and $mc_{-1}(a(\mathfrak{p}))^{k+1} = 0$, showing nilpotency in the mod $p$ reduction.
• When $\mu = n_1/n_2$ with $p \nmid n_1n_2$, it holds that $c_{\mu-1}(a(\mathfrak{p}))^{k+2} = 0$ and $mc_{\mu-1}(a(\mathfrak{p}))^{k+2} = 0$, confirming nilpotency under these conditions.
• The Grothendieck-Katz $p$-curvature conjecture holds for the system $D(L_n, \mathbf{a}, \mu)$, as shown via the connectivity of the motivic Galois group of the fiber.
• The system $D(L_n, \mathbf{a}, \mu)$ is globally nilpotent of rank 3, as established by Theorem 7.9 and Lemma 7.11.