Classification of rigid irregular G2‐connections

Abstract

Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured P 1 C having differential Galois group G2, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pullbacks we construct families of G2-connections which are not of these types.

Euphotic representations and rigid automorphic data

“…The starting point of our work is a new rigid G 2 -connection on P 1 \{0, ∞} found by the first-named author [11]. We looked for the automorphic representation corresponding to the -adic counterpart of that G 2 -connection, and arrived at the notion of euphotic automorphic data in general.…”

Euphotic representations and rigid automorphic data

“…5.1. The rigid connection from [Jak20]. The motivating example for our construction of rigid automorphic data is a certain rigid irregular G 2 -connection discovered by the first-named author.…”

“…The starting point of our work is a new rigid G 2 -connection on P 1 \{0, ∞} found by the first-named author [Jak20]. We looked for the automorphic representation corresponding to the ℓ-adic counterpart of that G 2 -connection, and arrived at the notion of euphotic automorphic data in general.…”

Wildly ramified rigid G2$G_2$‐local systems

“…One of the main ingredients of the classification in [9] is a classic result of Levelt-Turrittin for formal connections which allows to decompose any such connection into a direct sum of objects of the form…”

“…There are a lot of similarities and analogies in both settings, but unfortunately not everything translates directly from one to the other. The goal of this article is to introduce the necessary tools and methods to transfer the classification of [9] to the arithmetic setting.…”