Hypergeometric local systems over $\mathbb{Q}$ with Hodge vector $(1,1,1,1)$

TL;DR

This paper classifies all 47 irreducible rank-4 hypergeometric local systems over Q with a specific Hodge structure, linking them to families of threefolds and analyzing their geometric and arithmetic properties.

Contribution

It provides a complete classification of these local systems and connects each to geometric families, expanding understanding of their monodromy and Hodge structures.

Findings

47 classified local systems with specific Hodge vector

All local systems associated to families of threefolds

Analysis of geometry and arithmetic at conifold points

Abstract

We consider all irreducible rank-4 hypergeometric local systems defined over $\mathbb{Q}$ that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point.