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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?
This here seems to be the entry point:
The completed zeta function is
$\hat \zeta(s) \coloneqq \pi^{-s/2}\Gamma(s/2)\zeta(s) \,.$This has an integral-representation in terms of the theta function
$\theta(x)\coloneqq \underset{n \in \mathbb{Z}}{\sum} \exp(- \pi n ^2 x)$in the form
$\hat \zeta(s) = \int_0^\infty (\theta(x^2)-1) x^s \frac{d x}{x} \,.$Link for Nohara’s paper http://geoquant.mi.ras.ru/nohara.pdf at theta function does not work.
Thanks. Not sure what the problem was, now it works again.
Not Found
The requested URL /nohara.pdf was not found on this server.
Hm, I don’t understand what’s going on. It worked a minute ago. I google for the title “Independence of polarization in geometric quantization” get the link, it works from the Google pages, it works the first time after including it into the nLab page, and then it stops working?? I must be doing something weird without noticing.
In section 2.2 of
is reviewed how the relation in #2 above generalizes to general number fields, expressing now the Dedekind zeta function as an integral over a Hecke theta function.
By private email, Ivan Fesenko has been giving me invaluable information. Based on his explanation I have added a comment here on how to understand how the (Jacobi) theta-function gives rise to the (Riemann) zeta function by means their adelic integral-expressions.
One fact which I did not appreciate before is that, by the adelic integral-representation of the Riemann zeta function it is effectively a Gaussian integral on the space of adeles. But under the geometric Langlands analogy this space of adeles translates into the space of gauge fields of a gauge theory. So via its adelic integral representation the Riemann zeta function looks on the nose like a Wick-rotated path integral in gauge theory. (!)
I am aware that there are many (actual and suggested) relations between zeta functions and (path integral) quantization, but this particularly suggestive one I had not been aware of before. Is it just me?
Wick rotated is formally equivalent to the language of statistical mechanics and partition function, Connes and his collaborators extensively studied partition functions which yield various zeta functions and n some works adeles play a prominent role. Maybe you can find the case which you are excited about in their terms ?
To be more explicit, the integral formula that I mean is this one here.
Does Connes ever talk about this adelic integral?
On page 68 of Fesenko 10 there is a brief remark that something appearing in the adelic integration theory is “dual” to what appears in Connes’ theory.
I am not good enough to spot it, though I would be surprised if it were not there at least implicitly. Well, you should ask them. In writing it is difficult to extract exactly the same (maybe it is implicitly there) as his primary concern is to have adele classes as forming geometric space where zeros and some sums involving zeros of zeta function have spectral representation (concrete formulas realized as Lefschetz formulas and alike). He also uses Schwarz distributions there, corresponding Hecke algebras and other characters in your wider story.
By the way, the paragraph at Riemann zeta function
This adelic integral-method generalizes to Dedekind zeta functions for any algebraic number field. This is due to (Tate 50), highlighted in (Goldfeld-Hundley 11, Remark (1) by Ivan Fesenko).
is much of an understatement. Tate’s 1950 thesis is not in generality of Dedekind zeta function only but in wider generality of “Hecke zeta function”s what includes more cases.
I added a redirect (and few similar changes like links to wikipedia) to ring of adeles as they were originally (by Hecke) called valuation vectors. According to wikipedia, the word idèle comes from the 1936 Chevalley’s thesis who invented the group of idèles as (in the words of Tate) a “refinement of the ideal (class) group”. The adeles appeared even later by analogy and renaming of the earlier notion of valuation vectors, as used by Hecke and followers, say in famous
Okay, thanks!
I started a rough stub Iwasawa-Tate theory with the redirect Tate’s thesis. It is much overlapping with adelic integration but I think it should be distinct. First of all, the adelic integration is also studied in connection to the motivic integration and related questions of model theory. Second Iwasawa groups are studied in their own right as a parallel of a theory of reductive Lie groups and in relation to question of representation theory. Does both subjects have something what is not in their proper intersection, though the heart of the subject is belonging to the both titles. Of course, I have very vague impressions of the field and this attitude is open to critique.
Thanks, Zoran, for starting these entries! I have made a few more links come out at (and towards) Tate’s thesis
I am beginning to think that the most remarkable aspect of the adelic integration formula for the Riemann zeta function is the relation to the Jacobi theta function as discussed here. The crucial step there is that the Jacobi theta function is itself an integral of the form
$\begin{aligned} \int_{\mathbb{Q}^\times}\exp(-S(x n)) d n & = \sum_{n\in \mathbb{Z}^\times} exp(-\pi n^2 x_\infty^2) \\ & = \theta(x_\infty^2)-1 \end{aligned}$where I have suggestively written as “$\exp(-S(-))$” what in the literature is traditionally just called “h(-)” or “f(-)”.
Because notice the big analogy that we are after: the theta function (as discussed there) we are really to think of as a prequantum line bundle on a phase space of Chern-Simons theory, and moreover over a complex curve $\Sigma$ this line bundle is effectively just the transgression of the Chern-Simons 3-bundle to the mapping space out of $\Sigma$. That Chern-Simons 3-bundle in turn is a differential incarnation of the cup square intersection pairing.
Hence in total this means that on general analogy grounds we are to expect arithmetic theta functions to arise as integrals over Chern-Simons fields of a secondary cup-square action. Now by the Weil uniformization theorem and the fiunction field analogy, the space of ideles is to be thought of as just that, cocycles for $GL_1$ gauge fields “on Spec(Z)”.
Now I don’t see this exactly yet, but somehow the above expression for $\theta$ looks like it may be going in this direction…
Also, when regarding this $\exp(-S(-))$ as an action functional on gauge fields, then the gauge fixed path integral is indeed not over the space of field cocycles $\mathbb{A}^\times_{\mathbb{Q}}$ but over the quotient of that by the action of the gauge group, which here is $\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times / \mathbb{A}_{\mathbb{Z}}^\times$.
So in fact then the Jacobi theta function in adelic integral representation looks rather like the gauge orbit averaging of the action functional, while the remaing integral that gives the Riemann zeta function would be the remaining path integral proper. Hm…
The definition of theta-function $\theta^\ast_f$ as in Garrett 11, 1.9 is – in view of the interpretation of $\mathbb{A}_{\mathbb{Q}}^{\times}$ as the space of line bundles and of $\mathbb{Q}^\times \times \mathbb{A}_{\mathbb{Z}}^\times$ as the group of gauge transformations acting on that – just the “group averaging” of the function $f$, making it gauge invariant with respect to the $\mathbb{Q}^\times$-factor of the gauge group.
Hmm….
I put some comments at zeta function of an elliptic differential operator – Analogy with number-theoretic zeta functions.
It seems to me that this pretty much clarifies the question that I had at the beginning of this thread (and it’s of course in the literature, though not advertised as widely as it ought to be).
Interesting, the same Bost (coauthor of Vafa and Moore in the cited papers) as in the Bost-Connes model.
Given the table, I am getting a different geometric intuition as to what Langlands correspondence is about:
Consider here the association of automorphic representations to Galois representations as just an intermedediate step, and regard the further association of the automorphic L-function to that as the key step. Then Langlands correspondence is about sending Galois representations to L-functions.
But geometrically and by the table, Galois representations are flat connections $A$ and L-functions are regularized traces of Dirac operators $D_A$ twisted by flat connections.
So from this perspective it looks as if the Langlands correspondence is geometrically simply the assignment $A \mapsto Tr_{reg}( D_A^{-s})$.
(?!)
Is the following true: I imagine that
twisting a Dirac operator $D$ by a (flat) connection $A$ and then producing the zeta function of the Laplace operator $(D_A)^2$ is analogous to
twisting a number-theoretic zeta function by introducing a non-trivial Dirichlet character/automorphic representation.
Is that so? Might anyone have a pointer to a reference that would illuminate this relation?
One place where I see the similarity made more manifest between a) Artin L-functions twisted by Galois representations and b) zeta functions of Dirac operators twisted by flat connections is here.
made this an MO question here
MathOverfow: what-is-a-path-in-k-theory-space
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