started to collect some references at *Riemann hypothesis and physics*. But just a puny start so far, have to quit now.

started *Galois cohomology*

added the actual definition at *multiple zeta values* and added a paragraph relating to *motivic multiple zeta values*.

Also added to *motives in physics* a paragraph more explicitly mentioning the use of motivic multiple zeta functions for simplifying combinatorics of scattering amplitudes.

(prompted by this PO discussion)

]]>created *genus of a number field*

Added it to the function field analogy –table. Accordig to the footnote on the first page of Mazur-Wiles 83 the definition of this in Weil 39 is the origin of the function field analogy.

]]>have added a tad more to the Properties-section at *Riemann zeta function*.

needed to point to *ring of integers* of a number field. The term used to redirect just to *integers*. I have split it off now with a minimum of content. Have to rush off now.

just out of a whim, I expanded a little the text at *Fermat curve*

Added to *F1* a section *on Borger’s absolute geometry* and then split it off as a stand-alone entry (minimal as it is) *Borger’s absolute geometry*.

We have had our share of the debate of whether $Spec(\mathbb{Z})$ is really usefully analogous to a 3-manifold, and of how the $Spec(\mathbb{F}_p)$-s inside it then are analogous to knots in a 3-manifold.

Here is a thought (maybe this was voiced before and I am just being really slow, please bear with me):

things would seem to fall into place much better if we thought of the $Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$ not as analogous to knots, but as analogous to the prime geodesics inside a hyperbolic 3-manifold.

With this and its generalization to function fields, then the analogy between the Selberg zeta function for 3-manifolds and the Artin L-function (pointed out here) would become even better: in both cases we’d have the infinite product over all prime geodesics of, essentially, the determinant of the monodromy of the given flat connection over that geodesic.

Also, thinking of the $Spec(\mathbb{F}_p)$ not as knots but as prime geodesics removes all the awkward aspects of the former interpretation, such as why on earth one would be required to consider all these knots at once (which does not fit the analogy with knots in CS theory). Of course the prime geodesics would also be knots, technically, but I am talking here about the difference between thinking of them playing the conceptual role of the knots in CS theory (which are things we choose at will to build observables) and the prime geodesics, which are given to us by the gods as a means to compute the perturbative CS path integral.

Finally, there is of course much support from other directions of an analogy between prime geodesics and prime numbers (asymptotics etc.).

So it would seem to make much sense.

]]>created a bare minimum at *Diophantine equation*, just for completeness.

Also made *Diophantine geometry* a redirect to *arithmetic geometry* and added there one line saying way.

I saw that even Wikipedia doesn’t know these links, so I noted them down at *Weil conjecture on Tamagawa numbers*:

The announced proof by Jacob Lurie and Dennis Gaitsgory via nonabelian Poincaré duality of the Weil conjecture on Tamagawa numbers was announced in

- Jacob Lurie,
*Tamagawa Numbers via Nonabelian Poincaré Duality*, talk at FRG Chern-Simons workshop, Jan. 15-17, 2011

and details are at

- Jacob Lurie,
*Tamagawa Numbers via Nonabelian Poincare Duality (282y)*, lecture notes, 2014 (web)

finally created a minimum at *Dirichlet theta function*, cross-linked with *Dirichlet character* and *Dirichlet L-function* and added it to the table (bottom left entry)

(I have gotten a funny problem with my Opera browser having trouble loading nLab pages. Something makes it choke. For instance when I try to edit a page it tends to show me a blank screen, but when I then go to edit the same page with another browser, then that informs me that the page is locked, so Opera did get to that point, but then got stuck. This happens since the last few days. I tried clearing caches, but it didn’t seem to help. Hm. )

]]>created a table-for-inclusion and included it into the relevant entries:

*zeta-functions and eta-functions and theta-functions and L-functions – table*

Presently it looks like this:

The main statement of this analogy is discussed for instance on the first pages of

- Ken Richardson,
*Introduction to the Eta-invariant*(pdf)

I have added some paragraphs at *eta invariant*, accordingly.

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*itselfmore 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.

In some thread here (which I seem to have lost) there was the open question of whether the Selberg zeta function is indeed the zeta function of the corresponding Laplace operator. The answer is of course Yes, I have added the following paragraph to *zeta function of a Riemann surface*:

That the Selberg zeta function is indeed proportional to the zeta function of a Laplace operator is due to (D’Hoker-Phong 86, Sarnak 87), and that it is similarly related to the eta function of a Dirac operator on the given Riemann surface/hyperbolic manifold goes back to (Milson 78), with further development including (Park 01). For review of the literature on this relation see also the beginning of (Friedman 06).

(the links will only work from within the entry)

]]>gave *Langlands correspondence* an actual Idea-section.

(Am in a rush and on a horrible wifi connection. Need to proof-read and add more links later.)

]]>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 $K$

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*.

started some minimum at *Bost-Connes system*.

Hm, it seems that the statement is that that partition function of the BC-system

$\beta \mapsto Tr(\exp(- \beta H_{BostConnes}))$is the Riemann zeta function. But by the pertinent analogies the zeta functions are not supposed to *equal* partition functions, but to be related to them by the transformation

Hm.

]]>added to *zeta function of an elliptic differential operator* also some first minimum comments on the functional determinant and zeta-function regularization

created some bare minimum at *mod p Whitehead theorem*

started a note at *local-global principle*.

Need to interrupt now. This clearly can be extended indefinitely…

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