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

added pointer to

- Tom Lovering,
*Etale cohomology and Galois Representations*, 2012 (pdf)

for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to *local system* and maybe elsewhere.

Recorded the main idea at *special values of L-functions*.

have expanded the Idea-section at *L-function* in an attempt to transport some actual idea. The main addition are these paragraphs:

The most canonically defined class of examples of L-functions are the *Artin L-functions* defined for any Galois representation $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the Euler products of, essentially, characteristic polynomials of all the Frobenius homomorphisms acting via $\sigma$.

Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:

for 1-dimensional Galois representations $\sigma$ (hence for $n = 1$) Artin reciprocity produces for each $\sigma$ a Dirichlet character, or more generally a Hecke character $\chi$, and therefrom is built a

*Dirichlet L-function*or*Hecke L-function*$L_\chi$, respectively, which equals the corresponding Artin L-function $L_\sigma$;for general $n$-dimensional Galois representations $\sigma$ the conjecture of Langlands correspondence states that there is an automorphic representation $\pi$ corresponding to $\sigma$ and an

*automorphic L-function*$L_\pi$ built from that, which equalso the Artin L-function $L_\sigma$.

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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.

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)

]]>Finally added to *fracture theorem* the basic statement of the “arithmetic fracture square”, hence the following discussion.

The number theoretic statement is the following:

+– {: .num_prop #ArithmeticFractureSquare}

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$=–

In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).

+– {: .num_remark}

Under the function field analogy we may think of

$Spec(\mathbb{Z})$ as an arithmetic curve over F1;

$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;

$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in number theory, see for instance thr following discussions:

=–

]]>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.

]]>finally added some pointers at *Spec(Z) – As a 3-dimensional space containing knots* and cross-linked a bit.

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

created *arithmetic jet space*, so far only highlighting the statement that at prime $p$ these are $X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p)$ (regarded so in Borger’s absolute geometry by applying the Witt ring construction $(W_n)_\ast$ to it).

This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with $Spec(\mathbb{Z}_p)$ regarded as the $p$th abstract formal disk.

Well, or at least this is what Buium defines. Borger instead calls $(W_n)_\ast$ itself already the arithmetic jet space functor. I am not sure yet if I follow that.

I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality $\flat$ is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, $\flat$ canonically sits in a fracture suare together with the “cohesive rationalization” operation $[\Pi_{dR}(-),-]$ and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base $Et(Spec(\mathbb{F}_1))$ that makes the cohesive fracture square reproduce the arithmetic one.

If Borger’s absolute direct image were base change to $Spec(\mathbb{Z}_p)$ followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for $Et(Spec(Z))$, fine, but why this one? Need to further think about it.)

]]>started a stub for *moduli space of Calabi-Yau spaces*. Nothing really there yet, except some references and some cross-links.

gave *automorphic L-function* a minimum of an Idea-section, presently it reads as follows:

An *automorphic L-function* $L_\pi$ is an L-function built from an automorphic representation $\pi$, in nonabelian generalization of how a Dirichlet L-function $L_\chi$ is associated to a Dirichlet character $\chi$ (which is an automorphic form on the (abelian) idele group).

In analogy to how Artin reciprocity implies that to every 1-dimensional Galois representation $\sigma$ there is a Dirichlet character $\chi$ such that the Artin L-function $L_\sigma$ equals the Dirichlet L-function $L_\chi$, so the conjectured Langlands correspondence says that to every $n$-dimensional Galois representation $\sigma$ there is an automorphic representation $\pi$ such that the automorphic L-function $L_\pi$ equals the Artin L-function $L_\sigma$.

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A little bit of moonshine-style observations:

My last round of arithmetic geometry back then, after visiting Minhyong Kim last year, culminated in the observation (here) that the true differential-geometric analog of the Artin L-function for a given Galois representation is the perturbative Chern-Simons invariant of a flat connection on a hyperbolic manifold expressed as a combination of Selberg/Ruelle zeta functions.

Now the combination of Chern-Simons invariants and hyperbolic manifolds appears prominently in analytically continued Chern-Simons theory, where the imaginary part of the CS-action is given by volumes of hyperbolic manifolds (e.g. Zickert 07). (Thanks to Hisham Sati for highlighting this.)

Next, this formal complex combination $CS(A) + i vol$ also appears as the contribution of membrane instantons in M-theory, it’s the contribution of the Polyakov action functional of a membrane wrapped on a 3-cycle, if we read $CS(A) = C$ the contribution of the supergravity C-field.

Not sure how it all hangs together, but there might be some interesting relation here…

]]>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.

]]>I have started some bare minimum in entries

and cross-linked a bit.

]]>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.

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. )

]]>started some minimum at *vacuum amplitude*. Briefly mentioned relation to a) generating functionals for correlators and b) to zeta functions and c) to expected evanishing in supersymmetric theories

Remarked that in view of b) and c) one is tempted to expect some relation between 1-loop vacuum amplitudes of supersymmetric field/string theories with the Riemann hypothesis. Added a pointer to the article ACER 11 which seems to find just that.

If anyone has further pointers to literature relating vanishing of susy 1-loop vacuum amplitudes and (generalized) Riemann hypotheses, please drop me a note.

]]>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.

What is an *Artin L-function*, conceptually? It must be something really obvious:

The Frobenius homomorphism is to be thought of as a element of the fundamental group and its action via a Galois representation as a holonomy/monodromy around that element.

Hence Artin L-functions are simply (products of) characteristic polynomials of monodromies.

Googling “monodromy characteristic polynomial ” yields… the *Alexander polynomial*.

Check out this sentence from the introduction of Stallings 87, it is about knots but reads just like the definition of the Artin L-function:

For a fiber surface $T$, the translation of the fibre around the base-space circle determines an element in the mapping-class group of $T$, a homeomorphism $h\colon T \to T$ well defined up to isotopy; this element is called the

holonomyof the fiber surface; theAlexander polynomialis the characteristic polynomial of the map the holonomy induces on $H_1(T)$.

This relation/analogy must be know, however Google doesn’t give further hints.

]]>I am beginning to think that all these things are secretly the same:

Langlands’ conjecture 3;

the modular functor, hence quantization of 3dCS/2dWZW.

Read number-theoretic Langlands from the CS/WZW-theoretic perspective (using this and this) then one has: on the collection of flat connections (Galois representations) we assign the theta functions which are the partition functions (essentially: L-functions) with “source fields” these flat connections. That’s the definition of the conformal blocks.

But that’s also just the interpretation of equivariant elliptic cohomology in QFT language, which is a refined picture of the modular functor.

From this perspective Langlands’s “functoriality” is Lurie’s “global equivariance”, the fact that the construction is natural in the gauge group.

In this picture the automorphic forms don’t necessarily appear prominently, they rather seem a technical means to express the theta functions (their L-functions), analogous to how for writing down the standard modular functor it may be useful, but not necessary, to express the conformal blocks in certain preferred coordinates.

]]>I have a good understanding (or so it seems to me) of the conceptual role of theta functions in particular as far as they play a role in geometric quantization. Based on comments along these lines in the motivation section at *differential cohesion and idelic structure* an attentive reader kindly contacts me to ask or rather remark that this would seem to suggest that in the arithmetic context there appear interesting relations to L-functions.

This I need to learn about: what is the relation (in arithmetic(-geometry)) between theta-functions and L-functions?

What would be good sources to start reading?

]]>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.)

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