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This here to collect resources on the observation that – in view of pertinent arithmetic/differential-geometry analogies – an Artin L-function of a Galois representation looks like the zeta function of a Laplace operator of a Dirac operator twisted by a flat bundle.
I currently see this in the literature in three steps:
the Selberg zeta function, which is originally defined as some Euler product, is specifially equal to an Euler product of characteristic polynomials (just as the Artin L-function). This turns out to be due to Gangolli77 and Fried86, and I have collected these references now at Selberg zeta function – Analogy with Artin L-function with a cross-linking paragraph also at Artin L-function itself
more specifically, those characteristic polynomials are those of the monodromies/holonomies of the given group representation, regarded as a flat connection. This is prop. 6.3 in Bunke-Olbrich 94.
finally, that product over characteristic polynomials of monodromies is indeed the zeta function of the bundle-twisted Laplace operator. This is the main point in Bunke-Olbrich 94, somehow, but I still need to fiddle with extracting a more explicit version of this statement.
Is there an analogue of the Dedekind zeta function being the product over a complete set of irreducible unitary representations of the Galois group of corresponding Artin L-functions to the power of the dimension of the representation (e.g., here)?
So that’s the same as the Artin L-function for a particular induced representation (here).
Ah, thanks for highlighting that fact.
So, yes, as far as I am concerned, zeta functions are L-functions for a particular twist/background field. Typically I would say they are the L-functions for the trivial twist (trivial background field), but of course what that trivial twist really is may be subtle in the arithmetic context.
So that formula you point to says that the Dedekind zeta function is the Artin L-function for the “universal twist” namely for the group representation which is the tensor product of all irreps.
Oh, just edited #2 to add fact about single representation.
Thanks, excellent, so in this sense it is indeed the trivial representation, just “induced-up”.
Should add a pointer to an actual textbook proof..
That little wrinkle at end of #2 is for extensions which aren’t Galois. Can the Selberg side of the analogy make any sense of such things?
Yeah, something is missing. In fact the zeta function in Bunke-Olbrich 94 which has that nice Artin L-function feeling to it is apparently not the analytic continuation of $Tr(D^{-2s})$. Something more indirect must be going on.
Relevant MO discussion is here.
However, I’ll be offline now for a good bit.)
added a pointer to Murty-Murty 12 in the entry (here).
This all needs more attention. I am however really forced offline now. Hopefully back tomorrow.
If flat connections are thought as corresponding to Galois representations, what of the structure of the collection of the later? Are their irreducible flat connections, tensor products, etc.
Ah, I see Witten answers the first, irreducible flat connections here being those for which the only elements of the gauge group that commute with all monodromies are the central elements ±1.
Do we even have written out the relationship between Galois reps and flat connections? I see here:
It has been known for a long time there is a deep and mysterious analogy between Galois representations and local systems, cf. appendix of [Kat87],
and
As mentioned in the introduction, there is a deep analogy between Galois representation and flat connections. Guided by this analogy, Katz [Kat87] defined the differential Galois group, by employing the Tannakian structure on the category of connections.
Kat87 is On the calculation of some differential galois groups.
Hi,
sorry, I fell into an overly long and entirely offline weekend. Also, coming back there are three referee reports in the inbox which I need to look into now (funny how it goes).
When I have that out of the way I’ll get back to discussion here.
But one quick remark in reply to what you write above: the relation between Galois representations and flat connections is like this:
the absolute Galois group of a field $F$ is the fundamental group of $Spec(F)$;
flat 1-connections are equivalent to homomorphisms out of the fundamental groupoid into $\mathbf{B}G$, which in the connected case is equivalent to group homomorphisms from the fundamental group to $G$.
The nLab says this more or less in entries such as algebraic fundamental group and etale homotopy, but it could be said much more explicitly. The wikipedia page Étale fundamental group has some pertinent remarks.
(A subtlety for our discussion is that we don’t quite want the fundamental group of $Spec(F)$, but of $Spec(\mathcal{O}_F)$ (for $\mathcal{O}_F$ the ring of integers of $F$). Need to think about that.)
More in a little while, when I am back.
Given this 3-dimensional business, are there interesting flat 3-connections to be had?
I seem to remember asking Minhyong about this once (or at least about whether higher homotopy was important in number theory). You’d think that might have been during that cafe discussion on Kim on Fundamental Groups in Number Theory, but I can’t see it there. He’s written up some nice exposition of the importance of base points for fundamental groups in number theory, such as this MO answer.
I still don’t have a clear picture of what the claim re 3d actually is. The best picture is that which we collected references about at arithmetic topology, and there the 3d picture was more of a tool to produce a certain 2d picture.
Of course there are “interesting 3d connections” to the extent that there is interesting $\pi_1$ of a 3d manifold. Indeed, the whole story here seems to be very much that of Chern-Simons/WZW theory, where we start out with connections in 3d (the gauge fields of Chern-Simons) and then concentrate on the flat 2d connections (which form the “covariant phase space” of fields for any given surface). So it seems clear that there is some close interaction between 2d and 3d here. But while meanwhile I see rather clearly the arithmetic geometry versions of the 2d theory, I am not yet sure which phenomenon in number theory is genuinely 3d, in this sense. But I suppose it will become clear when we stare at it long enough.
regarding the subtlety mentiond in #12:
a great source here is, of course, Lenstra 85, where this appears in example 1.12. Have added a brief remark to this extent at Galois group – Properties – Relation to fundamental group
re #2:
I have tried to sort that out now at Artin L-function in brief subsections:
So of course the Dedekind zeta of some field $L$ IS just the Artin L-function for trivial twist:
$\zeta_L = L_{L, \sigma = 1} \,.$This is the “morally important” statement: L-functions are zeta functions which are twisted by the data of a flat connection.
On top of that we have the behaviour of Artin L-function on induced representations, and this implies on top of the above that if $L/K$ is a Galois extension then
$\zeta_L = L_{L, \sigma = 1} = L_{K, Ind_1^{Gal(L/K)}1} \,.$Aha, here is where I asked Minhyong about higher homotopy. He replies that there’s enough going on with $\pi_1$.
In the Leeds talk, Minhyong stresses the vital importance of thinking about basepoints properly, such as about definitions of fundamental groups in terms of fibre isomorphisms, they
…have been around at least since the 1960’s, but it is rather striking that variation of the base-point has not been really attended to until fairly recently. The primary impetus for a serious reassessment appears to have some from the interaction with the Hodge theory of the fundamental group.
This is covered also in the Cambridge talk.
I have added pointers to these talks here.
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