Non-abelian Abel's theorems and quaternionic rotation

TL;DR

This paper explores non-abelian Abel's theorems and quaternionic rotations as a low-tech approach to compute trace functions of l-adic sheaves over finite fields, bypassing traditional automorphic methods.

Contribution

It introduces the concept of multiplication laws on the Galois-representation side, serving as precursors to automorphic lifts, and provides a framework for computing trace functions without Hecke eigenvalues.

Findings

Proposes non-abelian Abel's theorems as an alternative computational tool.

Establishes connections between Galois representations and trace functions.

Offers identities that determine trace functions with prescribed ramification.

Abstract

In order to compute with $l$--adic sheaves or crystals on a line over $\mathbb{F} _q$ a low-technology alternative to the traditional computation with the Hecke operators on the automorphic side could be helpful. A program which has evolved over the years in our discussions with M. Kontsevich centers around the concept that, in the geometric case, there must exist certain multiplication laws on the Galois--representation side that could be thought of as precursors of the automorphic lifts: non-abelian Abel's theorems, and their restrictions to diagonal, Clausen identities. To a varying extent, they can determine the trace functions of $l$--adic sheaves or crystals with prescribed ramification without directly appealing to the Hecke--eigen property on the automorphic side.