Abstract
Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured $\mathbb{P}^1_{\mathbb{C}}$ having differential Galois group $G_2$, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types.
Highlights
Rigid local systems are local systems which are determined up to isomorphism by the conjugacy classes of their local monodromies. They arise as solution sheaves of certain regular singular differential equations, for example, the Gaussian hypergeometric equation
[8] Katz explains how to study rigid local systems using middle convolution. He proves that any irreducible rigid local system can be obtained from a local system of rank 1 by iterating middle convolution and twists with other local systems of rank 1
The above list exhausts all possible formal types of irreducible rigid irregular G2-connections on open subsets of P1 with slopes having numerator 1. This provides a classification of irreducible rigid connections with differential Galois group G2 with slopes of the desired shape, in particular providing the aforementioned non-hypergeometric examples of such systems
Summary
Rigid local systems are local systems which are determined up to isomorphism by the conjugacy classes of their local monodromies. The above list exhausts all possible formal types of irreducible rigid irregular G2-connections on open subsets of P1 with slopes having numerator 1 This provides a classification of irreducible rigid connections with differential Galois group G2 with slopes of the desired shape, in particular providing the aforementioned non-hypergeometric examples of such systems.
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