Calabi-Yau operators of degree two

TL;DR

This paper classifies Calabi-Yau operators of degree two, revealing a variety with ten components and identifying the two that are arithmetically significant, along with cataloging known operators.

Contribution

It provides a complete parametric description of the variety of degree-two Calabi-Yau operators and identifies the arithmetically interesting components.

Findings

The variety has ten irreducible components.

Only two components admit arithmetically interesting operators.

69 known fourth order Calabi-Yau operators of degree two are described.

Abstract

We show that the solutions to the equations defining the so-called Calabi-Yau condition for fourth order operators of degree two defines a variety that consists of ten irreducible components. These can be described completely in parametric form, but only two of the components seem to admit arithmetically interesting operators. We include a description of the 69 essentially distinct fourth order Calabi-Yau operators of degree two that are presently known to us.