The n-Category Café

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!

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 a lot MK! That’s really helpful! But could someone tell us the actual parameters? Dumb physicists only really care about the numbers.

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.