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I have added the following sentence to class number formula and to all the other entries that the sentence links to:
Given a number field $K$, the Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional the product of the regulator with the class number of $K$
$\underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.$In particular I have also created regulator of a number field and cross-linked it with Beilinson regulator, which I have renamed to higher regulator.
Are there any ideas about the ’meaning’ of the class field formula? There’s interest in $s = 1$ behaviour in number theory elsewhere, i.e., the BSD conjecture, the relation of which to the class field formula this MO question asks about. This abstract sees a path also to the Tamagawa number conjecture.
Not sure yet, but this is exactly what I am after, I would like to understand the QFT “meaning” of the Beilinson conjectures in view of the function field analogy.
Presently my rough picture is this:
a) the derivatives of special values of L-functions express determinant lines of Dirac operators on the given arithmetic curve (by the discusison here)
b) the Beilinson regulator is, as discussed there, essentially an arithmetic Chern character, hence after differential refinement this is a kind of generalized Chern-Simons action functional
c) the story of self-dual higher gauge theory says that the partition functions of the self-dual fields are determinant lines of Dirac operators and to be identified with the prequantum line bundles of a Chern-Simons theory.
That’s how I suppose it all hangs together. But I still need to sort out a bunch of details…
Interest in derivatives of the zeta function at $s= 1$ in arithmetic cases (as in class number formula), and at $s = 0$ in the case of functional determinants, is related via the functional equation? Is there even a functional equation in the case of the zeta function of a general elliptic differential operator?
Hmm, that relies on the relevant $\theta$ function being a certain way, so I guess not.
I suppose so, yes.
the Beilinson regulator is, as discussed there, essentially an arithmetic Chern character, hence after differential refinement this is a kind of generalized Chern-Simons action functional
Remnder: in my memory, Kontsevich has in 2004 related Chern-Simons action functional to the Soule regulator.
Thanks for the reminder, that definitely sounds like something I’d like to know about.
You had mentioned this before here. I guess I know of the articles that Soulé has on regulators. But I am unable to make Google give me anything regarding Kontsevich and Chern-Simons theory related to that.
Do you maybe happen to remember any further keywords or other hints that might help track down what that 2004 insight of Kontsevich was?
Also related is my question http://mathoverflow.net/questions/177413/bsd-leading-term-coefficient-in-terms-of-places-without-distinction, but I expressed it very badly. If I’d gotten a decent answer, I would have added it the body of notes Urs is developing.
Thanks, David, for this pointer.
While it doesn’t answer your question, that article by Bloch which somebody gave you on MO looks useful, I have added a pointer to it here.
The term isogeny crops up in Ben Wieland’s answer to David R.’s question and in answers to Ben’s question in turn.
In his ICM talk, Charles Rezk defines an isogeny as
Isogeny: finite flat homomorphism of group schemes.
Does this tally with our
A homomorphism between algebraic groups that is a surjection and has a finite kernel?
Faithfully flat morphisms are epimorphisms.
Do you maybe happen to remember any further keywords or other hints that might help track down what that 2004 insight of Kontsevich was?
I think I have been taking the notes from the talk. They are not easy to find, but at some point I should try (remind me next month, I have a couple of deadlines in next few days; though I will this week look at few most likely places, what is 20% chance to find now). Maybe also Herbert Gangl remembers the story, as it came from the study of higher algebraic K-theory of fields which they collaborated about at the time; he was also present at the lecture. Goncharov was not at the lecture but he knows the story from a personal communication and he understands the context.
@David C
well flat is morally ’all fibres the same’, so I guess surjective, since the fibre over the identity element is not empty. And a fix to what you wrote: flat morphisms that are epimorphisms are faithfully flat.
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