3 added reference to Toby Gee's comment

I am pretty sure that when different number theorists say "one of the main goals of number theory is to understand Gal(Q-bar/Q)" they may well mean different things.

One example of what someone might mean, which has been touched upon above, but which I'd like to stress (because it's what I mean whenever I start seminars with this generic-sounding statement!) is Clozel's conjectures in his Ann Arbor paper.

Langlands would conjecture that for any automorphic representation of GL_n over a number field K, there should be an associated n-dimensional complex representation of what people might now call the Langlands group of K (Langlands really conjectured something a little more precise: he conjectured that a certain category that he could not quite define, but whose objects he understood, should have the structure of a Tannakian category; I've tried to make this statement "concrete" in the above).

One of the problems with Langlands' conjecture was simply that it's not the sort of thing that can be checked by example, because no-one really knows an intrinsic construction of the Langlands group of K.

However, there are theorems in the literature about constructing Galois representations from automorphic representations, for example class field theory (which tells us what the abelianisation of this Langlands group should look like: ((adeles of K)^* / K_^* ) ) and Deligne-et-al's theorem attaching p-adic Galois representations to holomorphic cusp forms for GL_2 over Q.

Hang on a minute though---Langlands was conjecturing the existence of complex Galois representations of some group we don't have a definition of---how does Deligne's theorem fit into this? Here we have a p-adic representation (which wouldn't be continuous if you chose an isomorphism Q_p-bar = C and considered the induced complex representation) of a group that at least we have a definition of (Gal(Q-bar/Q) in Deligne's case). So in fact Deligne isn't proving a special instance of Langlands' conjecture, he's proving something else.

Clozel formulated precisely what that "something else" should be: he conjectured that to any algebraic automorphic representation of GL_n(adeles of K) there should be an attached p-adic Galois representation. I think that nowadays one would also conjecture that conversely given a geometric p-adic Galois representation (i.e. a representation Gal(K-bar/K)-->GL_n(Q_p-bar) which looks reasonable in some way that can be made precise) it should arise in this way. People like Emerton and Kisin are well on the way to proving this for K=Q and n=2.

So "proving Clozel's conjecture" would be one way of making precise "understanding Gal(Q-bar/Q)". What this would boil down to in this case would be understanding all the representations of Gal(Q-bar/Q) which were (a) taking values in Aut(V) with V a f.d. vector space over the p-adics, (b) unramified outside a finite set of primes (c) de Rham at p, and one would be understanding them in the sense that one would be parametrising the set of unramified conjugacy classes Frob_ell in terms of something totally different (automorphic representations, which are analytic gadgets).

The book of Harris and Taylor is to a large extent the state of the art nowadays (but the theory is currently moving so fast)fast) [EDIT: see Toby Gee's comment below]. However the base field K is essentially always either totally real or CM in this book, so in some sense the general case of Clozel's conjecture is still wide wide open.

Note finally that in some sense what Langlands conjectures is still open even for modular elliptic curves. If E is a modular elliptic curve over Q then the Galois representation naturally associated to E is its Tate module. But the representation of the Langlands group attached to E would be a representation to GL2(C), whose image would land in U2(C), the trace of Frob_p would be a sub p/sqrt(p), and (in the non-CM case) the conjugacy classes of the Frob_p's would be "evenly distributed in U(2)"---a statement which turns into the Sato-Tate conjecture when you unravel it. Note that Taylor et al proved the Sato-Tate conjecture but they certainly did not prove it by constructing this representation! Indeed it's in some sense an ill-defined question to construct this representation, because we don't know what the Langlands group is.

2 fixed grammar, improved formatting

I am pretty sure that when different number theorists say "one of the main goals of number theory is to understand Gal(Q-bar/Q)" they may well mean different things.

One example of what someone might mean, which has been touched upon above, but which I'd like to stress (because it's what I mean whenever I start seminars with this generic-sounding statement!) is Clozel's conjectures in his Ann Arbor paper.

Langlands would conjecture that for any automorphic representation of GL_n over a number field K, there should be an associated n-dimensional complex representation of what people might now call the Langlands group of K (Langlands really conjectured something a little more precise: he conjectured that a certain category that he could not quite define, but whose objects he understood, should have the structure of a Tannakian category; I've tried to make this statement "concrete" in the above).

One of the problems with Langlands' conjecture was simply that it's not the sort of thing that can be checked by example, because no-one really knows an intrinsic construction of the Langlands group of K.

However, there are theorems in the literature about constructing Galois representations from automorphic representations, for example class field theory (which tells us what the abelianisation of this Langlands group should look like: ((adeles of K)^* / K_^* ) ) and Deligne-et-al's theorem attaching p-adic Galois representations to holomorphic cusp forms for GL_2 over Q.

Hang on a minute though---Langlands was conjecturing the existence of complex Galois representations of some group we don't have a definition of---how does Deligne's theorem fit into this? Here we have a p-adic representation (which wouldn't be continuous if you chose an isomorphism Q_p-bar = C and considered the induced complex representation) of a group that at least we have a definition of (Gal(Q-bar/Q) in Deligne's case). So in fact Deligne isn't proving a special instance of Langlands' conjecture, he's proving something else.

Clozel formulated precisely what that "something else" should be: he conjectured that to any algebraic automorphic representation of GL_n(adeles of K) there should be an attached p-adic Galois representation. I think that nowadays one would also conjecture that conversely given a geometric p-adic Galois representation (i.e. a representation Gal(K-bar/K)-->GL_n(Q_p-bar) which looks reasonable in some way that can be made precise) it should arise in this way. People like Emerton and Kisin are well on the way to proving this for K=Q and n=2.

So "proving Clozel's conjecture" would be one way of making precise "understanding Gal(Q-bar/Q)". What this would boil down to in this case would be understanding all the representations of Gal(Q-bar/Q) which were (a) taking values in Aut(V) with V a f.d. vector space over the p-adics, (b) unramified outside a finite set of primes (c) de Rham at p, and one would be understanding them in the sense that one would be parametrising the set of unramified conjugacy classes Frob_ell in terms of something totally different (automorphic representations, which are analytic gadgets).

The book of Harris and Taylor is to a large extent the state of the art nowadays (but the theory is currently moving so fast). However the base field K is essentially always either totally real or CM in this book, so in some sense the general case of Clozel's conjecture is still wide wide open.

Note finally that in some sense what Langlands conjectures is still open even for modular elliptic curves. If E is a modular elliptic curve over Q then the Galois representation naturally associated to E is its Tate module. But the representation of the Langlands group attached to E would be a representation to GL2(C), whose image would land in U2(C), the trace of Frob_p would be a_p/sqrt(p)a sub p/sqrt(p), and (in the non-CM case) the distribution of the conjugacy classes of the Frob_p's would be "even"---a evenly distributed in U(2)"---a statement which is implies turns into the Sato-Tate conjecture when you unravel it. Note that Taylor et al proved the Sato-Tate conjecture but they certainly did not prove it by constructing this representation! Indeed it's in some sense an ill-defined question to construct this representation, because we don't know what the Langlands group is.

1

I am pretty sure that when different number theorists say "one of the main goals of number theory is to understand Gal(Q-bar/Q)" they may well mean different things.

One example of what someone might mean, which has been touched upon above, but which I'd like to stress (because it's what I mean whenever I start seminars with this generic-sounding statement!) is Clozel's conjectures in his Ann Arbor paper.

Langlands would conjecture that for any automorphic representation of GL_n over a number field K, there should be an associated n-dimensional complex representation of what people might now call the Langlands group of K (Langlands really conjectured something a little more precise: he conjectured that a certain category that he could not quite define, but whose objects he understood, should have the structure of a Tannakian category; I've tried to make this statement "concrete" in the above).

One of the problems with Langlands' conjecture was simply that it's not the sort of thing that can be checked by example, because no-one really knows an intrinsic construction of the Langlands group of K.

However, there are theorems in the literature about constructing Galois representations from automorphic representations, for example class field theory (which tells us what the abelianisation of this Langlands group should look like: ((adeles of K)^* / K_^* ) ) and Deligne-et-al's theorem attaching p-adic Galois representations to holomorphic cusp forms for GL_2 over Q.

Hang on a minute though---Langlands was conjecturing the existence of complex Galois representations of some group we don't have a definition of---how does Deligne's theorem fit into this? Here we have a p-adic representation (which wouldn't be continuous if you chose an isomorphism Q_p-bar = C and considered the induced complex representation) of a group that at least we have a definition of (Gal(Q-bar/Q) in Deligne's case). So in fact Deligne isn't proving a special instance of Langlands' conjecture, he's proving something else.

Clozel formulated precisely what that "something else" should be: he conjectured that to any algebraic automorphic representation of GL_n(adeles of K) there should be an attached p-adic Galois representation. I think that nowadays one would also conjecture that conversely given a geometric p-adic Galois representation (i.e. a representation Gal(K-bar/K)-->GL_n(Q_p-bar) which looks reasonable in some way that can be made precise) it should arise in this way. People like Emerton and Kisin are well on the way to proving this for K=Q and n=2.

So "proving Clozel's conjecture" would be one way of making precise "understanding Gal(Q-bar/Q)". What this would boil down to in this case would be understanding all the representations of Gal(Q-bar/Q) which were (a) taking values in Aut(V) with V a f.d. vector space over the p-adics, (b) unramified outside a finite set of primes (c) de Rham at p, and one would be understanding them in the sense that one would be parametrising the set of unramified conjugacy classes Frob_ell in terms of something totally different (automorphic representations, which are analytic gadgets).

The book of Harris and Taylor is to a large extent the state of the art nowadays (but the theory is currently moving so fast). However the base field K is essentially always either totally real or CM in this book, so in some sense the general case of Clozel's conjecture is still wide wide open.

Note finally that in some sense what Langlands conjectures is still open even for modular elliptic curves. If E is a modular elliptic curve over Q then the Galois representation naturally associated to E is its Tate module. But the representation of the Langlands group attached to E would be a representation to GL2(C), whose image would land in U2(C), the trace of Frob_p would be a_p/sqrt(p), and (in the non-CM case) the distribution of the conjugacy classes of the Frob_p's would be "even"---a statement which is implies the Sato-Tate conjecture when you unravel it. Note that Taylor et al proved the Sato-Tate conjecture but they certainly did not prove it by constructing this representation! Indeed it's in some sense an ill-defined question to construct this representation, because we don't know what the Langlands group is.