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Stream: theory: mathematics

Topic: Langlands reciprocity

John Baez (Feb 12 2025 at 17:39):

I'm finally knuckling down and trying to wrap my mind around Langlands reciprocity: like, what's the basic idea, ignoring as many technical details as possible, and why might we expect it to be correct?

Matthew Emerton, in a paper that's tragically not on the arXiv, argues that at one level Langlands reciprocity says every motivic L-function is also an automorphic L-function - and maybe vice versa, so that it deserves the name 'reciprocity'?

I understand motives reasonably well; very heuristically just as you can take $G$-sets for a group $G$, turn them into representations of $G$ called 'permutation representations' by composing them with the free functor $\mathsf{Set} \to \mathsf{Vect}$, and then decompose them into irreducible representations that aren't permutation representations, you can take an algebraic variety and use the magic of linear algebra to chop it into pieces called 'motives' that aren't algebraic varieties.

John Baez (Feb 12 2025 at 17:41):

Following this process, you can take the zeta function of an algebraic variety and write it as a product of functions associated to the motives, called motivic L-functions. Like zeta functions these are Dirichlet series, meaning they look like

$L(s) = \sum_{n = 1}^\infty a_n n^{-s}$

for some sequence of complex numbers $a_n$.

So roughly speaking, we've got an analogy:

set:vector space :: algebraic variety:motive :: zeta function of a variety:motivic L-function

John Baez (Feb 12 2025 at 17:51):

Automorphic L-functions are more mysterious to me, so I've been trying to learn them lately. Very roughly these are Dirichlet series somehow associated to 'automorphic representations'.

Langlands sometimes rather hand-wavily defines an automorphic representation as follows. Take a number field $F$. Let $\mathbb{A}_F$ be the ring of [[adeles]] for $F$. Then the group $GL(n,\mathbb{A}_F)$ contains $GL(n,F)$ so we can form the quotient space $GL(n,\mathbb{A}_F)/GL(n,F)$. The group $GL(n,\mathbb{A}_F)$ acts on this space in the usual way, and thus also on the complex-valued functions on this space. We get a representation of $GL(n,\mathbb{A}_F)$, and we can decompose it into irreducible pieces called automorphic representations.

Somehow he gets an L-function from such a thing; I find the formula on page 6 of his review paper mysterious.

John Baez (Feb 12 2025 at 17:58):

(In fact Langlands goes further and generalizes from $GL(n)$ to certain other groups, and then things get more complicated; I've been talking about what he called 'standard' automorphic representations.)

John Baez (Feb 12 2025 at 18:00):

Anyway, from this viewpoint, the reason why we might expect every motivic L-function to be an automorphic L-function is mainly that it's already known to be true in some important special cases, namely $GL(1)$ (which is called 'Artin reciprocity') and $GL(2)$ (which is called the 'modularity theorem').

John Baez (Feb 12 2025 at 18:00):

I would like a more conceptual reason for hoping that it's true!

The closest I got is this. There are a lot of conjectures about motives, and if they're all true motives are the same thing as representations of a certain group called the [[motivic Galois group]]. Similarly, automorphic representations seem to possibly be representations of some conjectural group called the 'Langlands group'.

Could it be that the motivic Galois group and the Langlands group are secretly isomorphic? I've seen someone wondering about this. But other people assured them that this was oversimplified. I can't find that conversation now, but there's some discussion of the idea here.

John Baez (Feb 12 2025 at 18:01):

At some point I temporarily gave up and decided to look into the [[geometric Langlands program]] a bit, since it's supposedly analogous but somewhat easier. Here I got a glimmering of a conceptual explanation, which I will talk about later. (Time to make breakfast!)

John Baez (Feb 13 2025 at 07:33):

Geometric Langlands is a variant of the Langlands program where we replace the number field $F$ by the field of meromorphic functions on a Riemann surface $X$. The analogy between the two kinds of field has been inspriring mathematicians for a long time, most notably Weil, so this is quite nice.

Edward Frenkel's introduction to geometric Langlands gives this formulation of Langlands reciprocity in this context:

Irreducible rank $n$ holomorphic vector bundles on $X$ equipped with a flat connection correspond to Hecke eigensheaves on the moduli stack of rank $n$ holomorphic vector bundles.

This is Theorem 3 on page 76.

John Baez (Feb 13 2025 at 07:36):

Presumably this is an equivalence of categories. I know what everything here means except "Hecke eigensheaf", so I need to learn what those are. Instead of "rank $n$ vector bundle equipped with flat connection", Frenkel says "rank $n$ local system", but he explains earlier that this is another way of thinking about the same thing.]

I think reading Frenkel's paper is the best way to make progress, but I'll stop here saying that the quoted formulation has a 'conceptual' feel to it even though I don't understand it! One interesting thing is that it claims a correspondence between 'things with a given structure' (vector bundles with a flat connection) and 'sheaves of some sort on the stack of things' (Hecke eigensheaves on the stack of vector bundles).

The funny level shift here reminds me a bit of the modularity theorem, which says very roughly that any elliptic curve over $\mathbb{Q}$ has, as a branched cover, a moduli space of elliptic curves.

I don't know what to make of that.

Also, from reading Frenkel's paper a bit, I know that 'Hecke eigensheaves' are related to the appearance of Hecke operators in the modularity theorem.

John Baez (Mar 01 2025 at 00:22):

By the way, I like Will Sawin's description of Langlands reciprocity:

There is a bijection between irreducible motives of rank $n$ and cuspidal algebraic representations of $GL_n$, characterized by each motive being sent to an automorphic form with the same L-function.

Simon Burton (Mar 03 2025 at 18:02):

I thought these cuspidal representations of $GL_n$ are not algebraic ... they are built from automorphic forms which are infinite sums...

John Baez (Mar 03 2025 at 18:15):

Hmm, good point. I don't know what Will Sawin meant by "algebraic" there.