# Base Change and the Artin Conjecture

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

Galois representations arise from many different problems in mathematics and encode large amounts of arithmetic data vital to the understanding of modern number theory. Associated to these representations, Artin constructed functions on the complex numbers called Artin L-functions, which encapsulate much of the information from the representation. These L-functions are at first only defined on a right half plane, but it has been proven that they extend to meromorphic functions (analytic functions that are allowed poles) on the whole plane, with a functional equation expressing a sort of symmetry possessed by the function. Artin conjectured further that in the case that the Galois representation is non-trivial and irreducible, the meromorphic continuation is actually analytic, i.e. has no poles. This is the Artin Conjecture, originally stated in 1923.

L-functions are not unique to the field of Galois representations, and in fact can be constructed in various other geometric and analytic scenarios. In particular, they have been constructed from Dirichlet characters and modular forms, analytic functions related to the groups GL(1) and GL(2) respectively. All of these L-functions have similar meromorphic continuation and functional equations, and in fact many classes of these Dirichlet and Hecke L-functions turn out to have analytic continuations.

The relationship between these two fields has been emerging over the last century, starting with Artin's abelian class field theory, which associates one dimensional Galois representations to Dirichlet characters. After that, the Modularity Theorem relates modular forms to elliptic curves, which form a special geometric source of 2-dimensional Galois representations. The proposed generalisation of both correspondences is the far reaching Langlands Program, which replaces Dirichlet characters and modular forms by automorphic representations, which are related to more general groups including GL(n). To these automorphic representations Langlands attached L-functions and proved their meromorphic continuation and functional equations. Furthermore, there is a notion of a cuspidal automorphic representation, which has analytic continuation of its L-function. The Langlands program now seeks a precise relationship between Galois representations and automorphic representations, which preserves the L-functions attached on either side. In fact, if non-trivial irreducible Galois representations can be associated with cuspidal automorphic representations, then this correspondence would prove that the L-functions of these Galois representations are in fact analytic, therefore proving the Artin Conjecture.

Following from the theory of Dirichlet L-functions, the case of one dimensional Galois representations has been fully settled, and so the next major aim is to work with GL(2). So far the most substantial progress in this direction came from Langlands in 1980, when he proved cyclic base change for GL(2). This result allowed him to construct automorphic representations corresponding to 2-dimensional Galois representations with solvable image, thus confirming the Artin conjecture for a large class of representations. For GL(n), Arthur and Clozel have proven that Galois representations attached to nilpotent field extensions correspond to automorphic representations, and so Artin's conjecture holds for these representations.

The aim of this project is to analyse the proofs of Langlands and Arthur-Clozel and attempt to make further progress on GL(n) for small n and deduce results on Galois representations and Artin's conjecture. This is a long outstanding problem that illuminates the deep connections between number theory and representation theory.

This project falls within the EPSRC Number Theory research area.

L-functions are not unique to the field of Galois representations, and in fact can be constructed in various other geometric and analytic scenarios. In particular, they have been constructed from Dirichlet characters and modular forms, analytic functions related to the groups GL(1) and GL(2) respectively. All of these L-functions have similar meromorphic continuation and functional equations, and in fact many classes of these Dirichlet and Hecke L-functions turn out to have analytic continuations.

The relationship between these two fields has been emerging over the last century, starting with Artin's abelian class field theory, which associates one dimensional Galois representations to Dirichlet characters. After that, the Modularity Theorem relates modular forms to elliptic curves, which form a special geometric source of 2-dimensional Galois representations. The proposed generalisation of both correspondences is the far reaching Langlands Program, which replaces Dirichlet characters and modular forms by automorphic representations, which are related to more general groups including GL(n). To these automorphic representations Langlands attached L-functions and proved their meromorphic continuation and functional equations. Furthermore, there is a notion of a cuspidal automorphic representation, which has analytic continuation of its L-function. The Langlands program now seeks a precise relationship between Galois representations and automorphic representations, which preserves the L-functions attached on either side. In fact, if non-trivial irreducible Galois representations can be associated with cuspidal automorphic representations, then this correspondence would prove that the L-functions of these Galois representations are in fact analytic, therefore proving the Artin Conjecture.

Following from the theory of Dirichlet L-functions, the case of one dimensional Galois representations has been fully settled, and so the next major aim is to work with GL(2). So far the most substantial progress in this direction came from Langlands in 1980, when he proved cyclic base change for GL(2). This result allowed him to construct automorphic representations corresponding to 2-dimensional Galois representations with solvable image, thus confirming the Artin conjecture for a large class of representations. For GL(n), Arthur and Clozel have proven that Galois representations attached to nilpotent field extensions correspond to automorphic representations, and so Artin's conjecture holds for these representations.

The aim of this project is to analyse the proofs of Langlands and Arthur-Clozel and attempt to make further progress on GL(n) for small n and deduce results on Galois representations and Artin's conjecture. This is a long outstanding problem that illuminates the deep connections between number theory and representation theory.

This project falls within the EPSRC Number Theory research area.

## People |
## ORCID iD |

Andrew Wiles (Primary Supervisor) | |

George Robinson (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2272745 | Studentship | EP/R513295/1 | 01/10/2019 | 31/03/2023 | George Robinson |