Artin's Conjecture and Elliptic Curves

TL;DR

This paper discusses Artin's conjecture on Galois representations and L-series, focusing on the one-dimensional case, historical developments, and the connection to modular elliptic curves and cusp forms.

Contribution

It provides a historical overview and explains the first verified example of the conjecture in the icosahedral case, linking Galois representations to elliptic curves and modular forms.

Findings

First verified icosahedral Galois representation in 1977

Connection between elliptic curves and cusp forms of weight 5

Illustration of the relationship between Galois representations and modular forms

Abstract

Artin conjectured that certain Galois representations should give rise to entire L-series. We give some history on the conjecture and motivation of why it should be true by discussing the one-dimensional case. The first known example to verify the conjecture in the icosahedral case did not surface until Buhler's work in 1977. We explain how this icosahedral representation is attached to a modular elliptic curve isogenous to its Galois conjugates, and then explain how it is associated to a cusp form of weight 5 with level prime to 5.