James Arthur

is a number theorist and representation theorist, working on the theory of automorphic representations (the Langlands program). He has proved many deep results on automorphic representations and their L-functions.

One (extremely pedestrian) way of understanding this theory, and some of Arthur's contributions to it, is this. Automorphic representations are some kind of analytic widgets, defined for reductive groups (e.g. the general linear group $GL(n)$) over number fields, which are a far-reaching generalization of modular forms. At least some of these are expected to be related to representations of Galois groups, in a functorial way. This last condition ("Langlands functoriality") roughly means that "natural" operations on Galois representations -- e.g. direct sums, tensor products, restricting to a subgroup, inducing up from a subgroup, etc -- should correspond to operations on automorphic representations; and one can hope to look for these purely in the automorphic world (even when, as in most cases, we can't prove anything about the link to Galois reps).

Arthur's work has confirmed the existence of many cases of these functoriality conjectures for automorphic representations. For instance his 1989 book with Clozel shows that automorphic representations of $GL(n)$ over a number field have operations corresponding to induction and restriction on the Galois side. This year he published a monograph on automorphic representations of orthogonal and symplectic groups, which shows (among a wealth of other things!) that there is a correspondence between these and auto reps of $GL(n)$ satisfying a suitable self-duality condition (analogous to the fact that a Galois representation into $GL(n)$ which is self-dual will land inside an orthogonal or symplectic group).

James Arthur

is a number theorist and representation theorist, working on the theory of automorphic representations (the Langlands program). He has proved many deep results on automorphic representations and their L-functions.

One (extremely pedestrian) way of understanding this theory, and some of Arthur's contributions to it, is this. Automorphic representations are some kind of analytic widgets, defined for reductive groups (e.g. the general linear group $GL(n)$) over number fields, which are a far-reaching generalization of modular forms. At least some of these are expected to be related to representations of Galois groups, in a functorial way. This last condition ("Langlands functoriality") roughly means that "natural" operations on Galois representations -- e.g. direct sums, tensor products, restricting to a subgroup, inducing up from a subgroup, etc -- should correspond to operations on automorphic representations; and one can hope to look for these purely in the automorphic world (even when, as in most cases, we can't prove anything about the link to Galois reps).

Arthur's work has confirmed many cases of these functoriality conjectures for automorphic representations. For instance his 1989 book with Clozel shows that automorphic representations of $GL(n)$ over a number field have operations corresponding to induction and restriction on the Galois side. This year he published a monograph on automorphic representations of orthogonal and symplectic groups, which shows (among a wealth of other things!) that there is a correspondence between these and auto reps of $GL(n)$ satisfying a suitable self-duality condition (analogous to the fact that a Galois representation into $GL(n)$ which is self-dual will land inside an orthogonal or symplectic group).

James Arthur

is a number theorist and representation theorist, working on the theory of automorphic representations (the Langlands program). He has proved many deep results on automorphic representations and their L-functions.

One (extremely pedestrian) way of understanding this theory, and some of Arthur's contributions to it, is this. Automorphic representations are some kind of analytic widgets, defined for reductive groups (e.g. the general linear group $GL(n)$) over number fields, which are a far-reaching generalization of modular forms. At least some of these are expected to be related to representations of Galois groups, in a functorial way. This last condition ("Langlands functoriality") roughly means that "natural" operations on Galois representations -- e.g. direct sums, tensor products, restricting to a subgroup, inducing up from a subgroup, etc -- should correspond to operations on Galois representations; and one can hope to look for these purely in the automorphic world (even when, as in most cases, we can't prove anything about the link to Galois reps).

Arthur's work has confirmed the existence of many cases of these functoriality conjectures for automorphic representations. For instance his 1989 book with Clozel shows that automorphic representations of $GL(n)$ over a number field have operations corresponding to induction and restriction on the Galois side. This year he published a monograph on automorphic representations of orthogonal and symplectic groups, which shows (among a wealth of other things!) that there is a correspondence between these and auto reps of $GL(n)$ satisfying a suitable self-duality condition (analogous to the fact that a Galois representation into $GL(n)$ which is self-dual will land inside an orthogonal or symplectic group).

James Arthur

is a number theorist and representation theorist, working on the theory of automorphic representations (the Langlands program). He has proved many deep results on automorphic representations and their L-functions.

One (extremely pedestrian) way of understanding this theory, and some of Arthur's contributions to it, is this. Automorphic representations are some kind of analytic widgets, defined for reductive groups (e.g. the general linear group $GL(n)$) over number fields, which are a far-reaching generalization of modular forms. At least some of these are expected to be related to representations of Galois groups, in a functorial way. This last condition ("Langlands functoriality") roughly means that "natural" operations on Galois representations -- e.g. direct sums, tensor products, restricting to a subgroup, inducing up from a subgroup, etc -- should correspond to operations on automorphic representations; and one can hope to look for these purely in the automorphic world (even when, as in most cases, we can't prove anything about the link to Galois reps).

Arthur's work has confirmed the existence of many cases of these functoriality conjectures for automorphic representations. For instance his 1989 book with Clozel shows that automorphic representations of $GL(n)$ over a number field have operations corresponding to induction and restriction on the Galois side. This year he published a monograph on automorphic representations of orthogonal and symplectic groups, which shows (among a wealth of other things!) that there is a correspondence between these and auto reps of $GL(n)$ satisfying a suitable self-duality condition (analogous to the fact that a Galois representation into $GL(n)$ which is self-dual will land inside an orthogonal or symplectic group).