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The American Institute for Mathematics has a great gift for spreading press releases about discoveries so esoteric…

Damn, that’s blown our chances of an AIM workshop! I was looking forward to a $n$-category conference in the new Alhambra.

I hope I didn’t blow our chances — I certainly didn’t intend to! I’ll see the original Alhambra this summer, but I want to run a $n$-category conference in the air-conditioned, wireless-enabled California replica as soon as it becomes available.

(I’ll be visiting Pilar Carrasco in Granada from June 21st to 29th — she of ‘hypercrossed complex’ fame.)

Do you know what the function actually is? I couldn’t find this information on their website.

I’d love to write about this at length, but I know that I’d run out of wind if I tried. So maybe I will contribute just a few remarks about the big picture. On the one hand, I agree with John that the article has an overblown flavor. On the other, I can guess why the results are very exciting to some people. Also, when I say ‘overblown’ I hope it’s clear that I’m referring to the press-release version that appeared across the link. I have no idea about the original lecture. Furthermore, I’m far from an expert on the specific issues, while Andrew Booker is a really impressive young mathematician who can talk circles around me as far the analytic theory of L-functions is concerned.

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There is a sequence of inclusions

zeta functions of varieties $\subset$ motivic L-functions $\subset$ automorphic L-functions

with the first and second class of objects being of primary interest. The reason for this is the information about the arithmetic of varieties that they are supposed to encode. This expectation is manifest in

-The class number formula that causes invariants of number fields to appear in the residue of its zeta function;

-The conjecture of Birch and Swinnerton-Dyer, proposing that arithmetic invariants of an elliptic curve are encoded in orders and leading terms of its L-function;

-Conjectures of Deligne, Beilinson, and Bloch and Kato about values of L-functions of varieties and motives. These are essentially grand generalization of the first two cases.

There are many interesting theorems by now related to these conjectures, even though the general framework is still rather speculative. But in any case where one can prove anything, one first has to prove that the zeta or L-functions have reasonable analytic properties. The Langlands’ program proposes that the analytic properties should always be proved by equating the L-function of a motive with the L-function of an automorphic form. Although I’m not defining either here, you should think of a motivic L-function as a ‘natural factor’ of the zeta function of a variety, while the latter is a generalization of the zeta function of a number field. L-functions of automorphic forms, on the other hand, are like Mellin transformations of modular forms. Because automorphic forms are special functions living on groups, like $GL_n$ or $Sp_n$, one expects their ‘Mellin transforms’ to have reasonable analytic properties. But then, by identifying a motivic L-function coming from an arithmetic variety of interest with some suitable automorphic L-function, one hopes to get an analytic handle on the former. It’s a very strange sounding strategy, even bizarre in some ways, but one that has come to be generally accepted by now, and of which the theorem of Wiles is a celebrated example.

Once one has absorbed all that and boiled it down to the expectation

All motivic L-functions are automorphic L-functions

it’s hard to avoid the question

Which automorphic L-functions are motivic L-function?

There is a conjectural answer involving a rather technical notion of an’algebraic automorphic form.’
*Algebraicity* is a generalization of the property of Hecke eigenforms on SL_2 that says that after normalizing the leading coefficient to be 1, the field generated by the coefficients of their q-expansions lie inside a number field. Using that notion, the answer takes the form

motivic L-functions are exactly L-functions of algebraic automorphic forms which furthermore have a small additional list of properties.

If I understand the usage of that article correctly, when they write ‘transcendental’ L-function, they mean the L-function of an automorphic form that is not algebraic.

If you’ve followed my mumbo-jumbo up to here, it is natural to ask the question:

‘Hmm. Then transcendental L-functions cannot be motivic. So why is the result so exciting? I thought you said those weren’t so interesting?’

What I believe to be the canonical answer of the moment to this question is rather striking: At the moment, number theorists are interested in good ways to embed motivic L-functions into larger natural families. One thinks of this procedure as being analogous to deforming the number field itself. One would like to do this for a number of reasons, but the primary motivation revolves, alas, around the Riemann hypothesis. This would be hard to explain in any detail, but the idea is that the Riemann hypothesis for general varieties over finite fields relies strongly on having L-functions living in very coherent families coming from families of varieties over finite fields. In fact, L-functions are built from the cohomology of a varieties, and studying the monodromy action of a parameter space on the cohomology of a family is *the key point* in the proof of the Weil conjectures. For this any many other reasons, people seem to feel it would be nice to have coherent families of L-functions satisfying certain properties, into which motivic L-functions fit.

If you don’t know anything about the Weil conjectures, perhaps you can still appreciate a general powerful principle of algebraic geometry (in fact, all of mathematics):

*Even if you want to study a single object, study the ways in which that object can move inside a family.*

This is the idea of moduli spaces. Of course number fields themselves are quite rigid. So to think of varying a number field in a geometric way, you might end up considering other types of objects, after which the ‘geometricity’ of the variation appears only in the L-functions.

Incidentally, because the families in the proven case rely on the existence of a base-field $F_p$ over which everything lives and is varied, the expectation for a ‘monodromy theory of L-functions’ over number fields is rather intimately related to speculations about $F_1$.

Now to the new result itself. There are many kinds of automorphic forms with associated L-functions, but they are quite hard to construct explicitly once one moves to groups larger than $GL_2$ (which carries the usual modular forms). The Maass forms in particular, are real-analytic eigenfunctions of the Laplacian on such groups. Because there are certain non-compact extensions of Weyl’s volume formula on these groups, one knows the existence of many Maass forms. But till now, I don’t believe there has been a single explicit construction other than for $GL_2$. If I understood the blurb correctly, the lecture under discussion gives the construction of the L-function of a Maass form on $GL_3$. It also appears that they construct the L-function even without the automorphic form. This may seem bizarre, except for a technical but powerful fact that’s been known since Weil’s book on modular forms in the sixties: There are analytic criteria for identifying an Euler product as being the L-function of an automorphic form. These are the so-called `converse theorems’ to which my formal colleague Freydoon Shahidi made seminal contributions. So I suspect that they have some Euler product that’s supposed to satisfy the criteria of the converse theorems, either numerically or provably, and thereby, a relatively concrete automorphic L-function of an genuinely new sort. (One can typically read off the type of the automorphic form from the L-function. I presume this is how they know that their construction comes from a Maass form.)

From the perspective of people who would like to produce as many automorphic L-functions as possible that fit into the scheme of Riemann-hypothesis like problems, it must certainly be an exciting discovery. But the assertion that this takes us genuinely closer to the RH seems exaggerated.

Finally, I should point out that I myself am not terribly convinced that L-functions are as incredibly important as is commonly believed. (Even though I really wanted to be convinced for many years.) I once heard from Swinnerton-Dyer that when he and Birch were first formulating those famous conjectures, zeta and L-function-ology was not taken at all seriously by ‘the mainstream’. Chevalley was apparently famously opposed to their use in number theory. Of course the pendulum has swung completely the other way and you see people claiming the L-world to be world of essentially all of number theory. Claims of this type I don’t buy into. To get a bit of my thoughts on this issue you can read my lecture at the IHES summer school on motives. Parts of it are quite dense and certainly not up to the expository standards of this cafe. But the first and last sections may be entertaining. The original slides (and references) can be found here .

Thanks very much for this. You could offer it to AIM.

I myself am not terribly convinced that L-functions are as incredibly important as is commonly believed.

You say in the lecture you refer to that there is as yet no anabelian analogue of the L-function. So, if these were found, you would be happier with the idea of “most of the secrets of number theory” living in the ‘World of anabelian L’?

That might improve things a bit. The kind of non-abelian L is actually a very concrete sort of object. It should lie in some non-abelian algebra instead of being an ordinary function.

But my doubt runs a bit deeper than that. Even the applications of L-function theory that already exist are suggestive of a more fundamental formulation. Perhaps I’ll have the energy to write more later. It would be good for me to write about it. But for the moment, I’m suffering from a backlog of chores, and preparing for an impending trip to Bangalore.

Just to add one more cryptic remark, what I’m referring to as a ‘more fundamental formulation’ could well be very category-theoretic. In standard applications, the L-functions appear as objects controlling intertwining operators between various Galois representations.

More later.

Kontsevich made *here* some strange remarks on nonabelian L-Functions:

“It is reasonable to imagine that there is a big non-commutative L -function one of whose limiting cases is arithmetic while the other is topological string theory.”

Posted by:
Thomas Riepe on March 18, 2008 5:45 PM | Permalink
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Is there a way to view Kontsevich’s remark in the light of Minhyong’s comment above about intertwining operators?

I have no idea about that because I know far too little of both concepts. Let’s wish Minhyong that his health recovers soon and that he finds time to tell us!

Posted by:
Thomas Riepe on March 19, 2008 12:29 PM | Permalink
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Ok…with the…last vestiges…of my…strength…I’ll make…a few remarks… about Kontsevich :=)

What he defines in that paper is quite different from what I have in mind, although at this point, I wouldn’t risk describing it as ‘unrelated.’ At a vague level, his L-function has the nature of De Rham cohomology, while the one I have in mind will be more etale. But more importantly, the L-function he defines is

* a homological complex L-function of a non-commutative space *

That is, he starts with some non-commutative space $A$ and considers its periodic cyclic homology $H(A)$. If we assume that $A$ is of finite-type over $Z$, then we can tensor with each $p$-adic completion $Z_p$ and get a $Z_p$ cyclic homology $H(A\otimes Z_p)$. With further assumptions he conjectures that this admits a natural action of a Frobenius at p and uses the Frobenius to define a local L-factor as

$L_p(A,s)=\det(I-p^{-s}Fr_p|H(A\otimes Z_p))$

Although the discussion is very sketchy, this part is somewhat plausible, since cyclic homology is supposed to be like De Rham cohomology. When defining the local L-functions for usual varieties, one can use for each prime p the De Rham cohomology of the variety rather than the etale cohomology. (The fact that you can do this follows from the theory of crystalline cohomology.) In any case, one puts all those together to define

$L(A,s)=\prod_p L_p(A,s)$

Since he moves onto a discussion of the zeta function, I presume that the determinant occurring in the definition is already an alternating product of determinants. That is, for a usual variety, it should look like

$L_p(H^{odd},s)/L_p(H^{even},s)$

and he must be considering something similar.

Now, again for usual varieties, the eventual $L(A,s)$ has another expression

$L(A,s)=\prod_x (1-N(x)^{-s})^{-1}$

where the $x$ run over the maximal ideals of $A$ and $N(x)=|A/x|$. (Here I am being very sloppy and speaking about the case of an affine scheme, although I should be taking projective schemes to make the cohomological formula work out in a straightforward way. One also needs to be careful with factors corresponding to the finitely many $p$ of ‘bad reduction.’) This can also be written

$L(A,s)=\Sigma_z 1/N(z)^s$

where z now runs over the effective zero cycles, that is, formal linear combinations of closed points, on $Spec(A)$. If you look at it in this form, you see that one can vaguely think of it as being a sum over finite A-modules. This is what Kontsevich means at the end when he speaks of a sum over a category $C_A$ with certain weights. Now, as a sum over modules, it’s not hard to imagine something of that sort arising in string theory, where one would be summing over A or B branes, whichever corresponds to coherent sheaves. A weighted sum incorporating sums over zero cycles and sums over branes is what Kontsevich is conjecturing.

What I’m thinking about, on the other hand, is

* an L-function of homotopical nature associated to a commutative space.*

It would be an element of some non-commutative group algebra, not a function of a complex variable. As for relations to Kontsevich, I suppose one could make a wild guess of the following sort:

Given a variety $V$, the non-commutative homotopical L-function of V should be related to Kontsevich’s homological L-function for the group algebra of $\pi_1(V)$.

But I wouldn’t put it in italics.

Manifestations of zeta or L-functions as elements of group algebras is one of the striking lessons of * Iwasawa theory *. There, a prime $p$ is chosen which leads to $p$-adic $L$-functions. They are actually more fundamental than complex $L$-functions, being the objects controlling spaces of intertwining operators between Galois representations. However, there should be something more basic underlying all that.

After considering for a moment this morning what I wrote above, it seemed reasonable to summarize the main point as follows:

Kontsevich * does not go far enough * in his thoughts about non-commutative L-functions. After all the trouble of setting up the formalism of non-commutative spaces, it’s quite disappointing that the L-function of such an object is an ordinary function.

Perhaps the problem is with cyclic homology being an abelian object.

For the kind of L-function I have in mind, I think the closest example from physics is in Connes and Kreimer’s perturbative approach to quantum field theory. There, ‘the theory’ is a point in some unipotent group, while numbers come about by evaluating functions (Feynmann diagrams) at that point. Iwasawa-theoretic L-functions are very much like the point corresponding to the theory, while a specific L-value is like the amplitude corresponding to a diagram. I will write about this analogy at greater lengths some time when I have more energy.

If as you said

…a ‘more fundamental formulation’ could well be very category-theoretic,

is there any interest in your area in what Ross Street and Elango Panchadcharam are doing by formulating Mackey functors in an enriched category theoretic setting (slides)?

They point out that Mackey functors are used in Iwasawa theory.

Very well done. Thank you!

I depend on local experts, such as Emeritus Professor Tom Apostol at Caltech for details.

But I love your phrase: “… deforming the number field itself…”

It’s like a Math-analog of General Relativity describing things in terms of deformations of the space-time continuum itself…

What is “the monodromy action of a parameter space?”

As to L-functions not being so terribly important—maybe not, but it would be nice to understand how they fit together, all the same.

Posted by:
Tim Silverman on March 18, 2008 10:18 PM | Permalink
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“What is “the monodromy action of a parameter space?””

Deligne proved the Weil conjectures by proving them simultaniously for a family of varieties (a “Lefschetz pencil”). Such a family is parametrized by some scheme, so the cohomology groups of the single varieties of the family fit together to a local system over the paramerizing scheme. Then he investigated the action of the monodromy group of the parameter scheme on that local system. There exist some very readable expositions of the theme, esp. by *Katz*. If someone is curious, I would recommend to read Katz’ ICM 1970 article “The Regularity Theorem in Algebraic Geometry”.

The first hint that one should look for such monodromy representations came acc. to Katz from Swinnerton-Dyer’s estimates made from the family of elliptic curves over the “universal elliptic curve” mentioned in the modular thread.

Posted by:
Thomas Riepe on March 19, 2008 10:00 AM | Permalink
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Since Thomas answered the question nicely, I’ll just comment briefly on your remark. I hope it’s clear that I also regard $L$-functions as very, very important, in fact, an indispensable part of an education in number theory at the present time. It’s just that I find the importance credible :)

More seriously, I was just objecting to the perception of $L$-functions as all-encompassing. I’ve already pointed out that they do not directly contribute, in their present form, to Diophantine geometry of any variety that is not abelian. This is because of their homological nature. A second point is that in a world where all conjectures were theorems, the utility of $L$-functions for providing even abelian arithmetic geometric information would still be considerably more limited than casual discourse would have you believe. This depends very much on the situation of course. For example, the class number formula is the most efficient way to compute the class number of quadratic or cyclotomic fields. Thus, some abelian invariants of an arithmetic geometric space are very efficiently encoded in an L-function.

In the case of an elliptic curve $E$, however, even were we to know the most famous part of the BSD conjecture

$ord_{s=1}L(E,s)=rank E(Q)$

the formula would have very limited utility in computing the rank of $E(Q)$. On the other hand, there is another part of the conjecture

* the finiteness of the Tate-Shafarevich group of $E$ *

that would give an effective algorithm for computing the rational points of $E$. In fact, existing algorithms for elliptic curves just assume the conjecture and search for points, even though the algorithm terminates only if the conjecture is true. This $Sha(E)$ part of the conjecture has a priori nothing to do with $L$-functions. Curiously enough, however, in all cases where it’s actually proved, $L$-functions enter the argument in mysterious ways.

Hmm…

Thanks for these very helpful replies, Thomas and Minhyong!

MK, I understand better where you are coming from now, and it makes a lot more sense to me (not that I doubted your good sense before—I just wasn’t clear where the emphasis was).

Posted by:
Tim Silverman on March 20, 2008 8:11 PM | Permalink
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Thanks a lot MK! That’s really helpful! But could someone tell us the actual parameters? Dumb physicists only really care about the numbers.

Sorry, I have no details. I’ll write to Booker at some point and try to get something on this.

A description seems be be *here*. At least the title indicates that, I can’t open the link on this computer. BTW, his website offers a *korean vocabulary*! Conc. Maas forms: I think I once saw a description of them in a book by Gelbart on automorphic forms on adele groups.

Posted by:
Thomas Riepe on March 18, 2008 1:22 PM | Permalink
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I remember him telling me about this vocabulary and my asking him about the rationale, but I’ve forgotten now. I glanced into the paper you pointed to, which is actually just for Maass forms on $GL_2$. But the basic outline must be similar. It indicates that some of my guesses above about their work was not too accurate, but oh well. It looked elementary enough to be accessible to a wide readership. If you don’t have subscription to IMRN, it’s also available for free on the homepage of Akshay Venkatesh (another bright young number-theorist).

I’ve been meaning to invite Booker to London for a while anyways. I’ll do this as soon as I recover from flu, backlog, etc.

Andrew Booker came by today to speak at the London Number Theory Seminar. I don’t have much to add to my previous comments, since they seem to have been tolerably accurate. But I thought you might be amused by his abstract:

Andrew Booker (Bristol)

“Computing automorphic forms on GL(3)”

Abstract: My student, Ce Bian, announced the computation of a few “generic” rank 3 automorphic forms (meaning they are not lifts from lower rank examples) at the AIM workshop “Computing arithmetic spectra” in March. I will give a brief introduction to the theme of the workshop and describe Bian’s computations. I’ll also say a few words about the bewildering amount of attention that the work received subsequently.

I do not understand much of what is going on here, but as an outside observer who worries about correctness of real-number computation I would be grateful if someone can explain why I need not to worry that the result was obtained by solving numerically (using CPU floating point numbers) linear systems in 10,000 variables. How was correctness of results established? Is there an easy way to check correctness? At first sight it seems that what they did is at least as shaky as Hale’s result on Kepler conjecture. People attacked him for using computers without proving all the computations and programs were correct.

## Re: The World of L

The American Institute for Mathematics has a great gift for spreading press releases about discoveries so esoteric that most mathematicians have no idea what’s actually been discovered. We could use some publicity like that.

And just think how much more attention they’ll get after they rebuild the Alhambra!