Hypergeometric Local Systems and Parabolic Bundles

TL;DR

This work connects the Beukers–Heckman criterion for algebraicity of hypergeometric functions to the Mehta–Seshadri correspondence by constructing explicit stable parabolic bundles associated to unitary hypergeometric local systems. The key insight is that unitary monodromy is equivalent to the stability of a carefully built parabolic bundle $E_\star(\alpha,\beta)$, with interlacing of $\{e^{2\pi i\alpha_i}\}$ and $\{e^{2\pi i\beta_i}\}$ characterizing when stability holds. The paper provides explicit descriptions of the parabolic bundle in this regime and a new geometric proof of Beukers–Heckman’s finiteness criterion: when $\alpha_i,\beta_i$ are rational and interlacing holds for all Galois conjugates, the hypergeometric local system has finite monodromy. The results integrate rigidity (Katz) with parabolic stability to recast the Beukers–Heckman theorem in terms of unitary representations arising from parabolic data, offering a concrete, constructive framework for understanding finite monodromy in hypergeometric systems.

Abstract

Beukers and Heckman gave necessary and sufficient conditions for a hypergeometric function $_n F_{n-1}$ to be algebraic. We give a new proof of this theorem by passing through the Mehta-Seshadri correspondence. In particular, we explicitly write down the parabolic bundle corresponding to a unitary hypergeometric local system.

Theorems & Definitions (29)

• Theorem 1.1
• proof
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Remark 3.1
• Proposition 3.2