Artin’s conjecture and elliptic curves

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. In 1917, Erich Hecke (10) proved a series of results about certain characters which are now commonly referred to as Hecke characters; one corollary states that one-dimensional complex Galois representations give rise to entire L-series. He re- vealed, through a series of lectures (9) at Princeton's Institute for Advanced Study in the years that followed, the relationship between such representations as gen- eralizations of Dirichlet characters and modular forms as the eigenfunctions of a set of commuting self-adjoint operators. Many mathematicians were inspired by his ground-breaking insight and novel proof of the analytic continuation of the L-series. In the 1930's, Emil Artin (1) conjectured that a generalization of such a result should be true; that is, irreducible complex projective representations of finite Ga- lois groups should also give rise to entire L-series. He came to this conclusion after proving himself that both 3-dimensional and 4-dimensional representations of the simple group of order 60, the alternating group on five letters, might give rise to L-series with singularities. It is known, due to the insight of Robert Langlands (16) in the 1970's relating Hecke characters with Representation Theory, that in order to prove the conjecture it suffices to prove that such representations are associated to cusp forms. This conjecture has been the motivation for much study in both Algebraic and Analytic Number Theory ever since. In this paper, we present an elementary approach to Artin's Conjecture by con- sidering the problem over Q. We consider Dirichlet's theorem which preceeded Hecke's results, and sketch a proof by introducing theta series. We then introduce Langland's program to exhibit cusp forms. We conclude by studying a specific ex- ample which is associated to an elliptic curve. We assume in the final sections that the reader is somewhat familiar with the basic properties of elliptic curves.