Wildly Ramified Rigid $G_2$-Local Systems

TL;DR

This paper extends the classification of rigid $G_2$-local systems to wildly ramified cases with slopes having numerator 1, adapting methods from irregular connections to positive characteristic $\,\, ext{settings}.

Contribution

It provides the necessary results to classify wildly ramified rigid $G_2$-local systems with specific slope conditions, adapting previous methods to positive characteristic.

Findings

Classification of wildly ramified rigid $G_2$-local systems achieved.

Methods for invariants differ from previous irregular connection approaches.

Framework applicable to positive characteristic settings.

Abstract

In earlier work of the author rigid irregular connections with differential Galois group $G_2$ and whose slopes have numerator $1$ were classified and new rigid connections were constructed. The same construction can be carried out for $\ell$-adic local systems in the setting of positive characteristic. In this article we provide the results that are needed to obtain the classification of wildly ramified rigid $G_2$-local systems whose slopes have numerator $1$. The overall strategy of the classification is very similar but the methods needed to obtain some invariants differ.