Mathematics – Page 37 – E. Kowalski's blog

A tale of two dichotomies

In arithmetic and geometry, there is a well-known split between right-wingers who write

$G/\Gamma$

and left-wingers who prefer

$\Gamma\backslash G$

More insidious and mysterious is the deeper split between the inliners who write

$SL(2,\mathbf{Z})$

and the redoubtable subscripters who will only write

$SL_2(\mathbf{Z}).$

I’ve used almost exclusively the first (almost because I don’t think a paper referring to E(8,Z) could possibly be accepted), but I have no particular memory of why I started. Does anyone have an argument (in bad faith or otherwise) for either choice?

Kloosterman vs. Salié

I’ve just realized a rather obvious fact concerning the vague question at the end of my latest post on Kloostermania, which I’ll rephrase informally:

Can one guess with better than even chance that a graph like Kloostermania’s represents a Kloosterman or a Salié sum?

I had said that, for fixed p, just knowing the sum shouldn’t leave much room to make any choice except a random one. To make sense, the rule must be made more precise: you are given a bare real number, say

$S=-17.0711787748\ldots$

and you are told that it is either the value of a Kloosterman sum S(1,1;p) for some prime, or of a Salié sum T(1,-1;p) for some prime. You can win a drink of your choice by picking up the right one. What can you say? The reason it is hard to do better than heads-Kloosterman/tail-Salié is that you do not know p. If you knew the value of the prime, then you could compute the angle in [0,π] such that

$\frac{|S|}{2\sqrt{p}}=\cos(\theta),$

and use the fact that these are supposed to be distributed differently for Kloosterman and Salié sums. For instance, the probability that θ is(conjecturally) larger than 3π/4 is bigger for Salié sums (it is 1/4) than for Kloosterman sums (about 0.09…), and hence, finding an angle in this range would give a big hint that it comes from a Salié sum (but no certainty, of course).

And now for the really obvious point: if you can, in addition to knowing S, actually see the graph of the partial sums, then — of course — you know the prime: you just have to count the steps on the graph.

It seems that similar ideas should lead to a slightly better than average guess even without knowing p: the size of S gives a lower bound on p, and for each possible guess of p, we have a guess of either Salié or Kloosterman. Presumably, combining these should be possible to get a small gain on pure chance…

(Note: readers are welcome to make a guess concerning the value above…)

Roman dodecahedron

The platonic solids are of course quintessentially Greek (although a claim to their discovery has apparently been staked on behalf of rugged Scots — who certainly play rugby better, not that this should influence priority disputes). I was therefore quite intrigued to see today, in the Roman Museum of the town of Avenches, a very beautiful Roman dodecahedron:

The decorations are quite interesting; note for instance that the holes in the faces are not all of the same size. The accompanying text mentioned that at least 60 such objects have been found in what was ancient Roman territories north of the Alps, and that their purpose (if any) is not known. This one was found in a private house (the approximate date is not mentioned, but the old Roman city of Aventicum apparently flourished mostly during the first Century).

WordPress mobile ads

This is rather off-topic, but maybe not all owners of WordPress-hosted blogs are aware of it: since a few days, the mobile version of their sites (which displays when accessed from a phone, or at least from an Android one) displays a small ad-widget by default. I’ve noticed this on Vieux Girondin, Quomodocumque, What’s New, T.Gowers’s blog, etc… Here’s what it looks like:

If this is not the desired behavior (and some of the ads I’ve seen might not be what the owners like to be associated with; the one above is not the worst…) it is probably easy to disable the plugin in charge of the mobile theme (but I can’t check; although my own blog is WordPress-powered, it is hosted at ETH and doesn’t have this plugin).
Also, unfortunately, I don’t know how to check the mobile version from a desktop or laptop computer (presumably one needs to tell the browser to pretend to be a mobile version, but I’ve never learnt how to do that…).

More Kloostermania news

Kloostermania fans can look up what the new version 0.15 does on the Kloostermania page…

In particular, I’ve added the possibility to display Salié sums instead of Kloosterman sums. Precisely, it shows

$T(1,-1;p)$

instead of T(1,1;p) because the values of the latter are not as interesting, due to the fact that the root s of

$X^2=4$

modulo a prime are rather simple to compute….

This leads to an intriguing game to play: can you tell the difference between the two types of sums?

First, one must take a prime congruent to 1 modulo 4 (otherwise the Salié sum is zero in that case, which Kloosterman sums never are) for the question to be interesting. Then, there is a kind of theoretical/conjectural answer if you are allowed to look at many instances of the two sums (i.e., you can start the slideshow and observe it for a long time — skipping primes which are 3 mod 4 — without changing the type of sums): their distribution is not the same (conjecturally)! Precisely, the angles θ_p of the Salié sums, for primes which are 1 modulo 4, defined by

$T(1,-1;p)=2\sqrt{p}\cos\theta_p$

are equidistributed on [0,π] (for the Lebesgue measure; this is the wonderful theorem of Duke, Friedlander and Iwaniec), whereas one expects those of Kloosterman sums to be distributed according to the Sato-Tate measure

$\frac{2}{\pi}\sin^2\theta d\theta.$

In particular, the Kloosterman sums should be more often “small”, in some sense, than the Salié sums since the density of the Sato-Tate measure vanishes at θ=0 (which corresponds to a maximal sum).

But what if you’re not allowed to start a long slideshow? For a fixed p, I don’t think one can expect to be able to guess more precisely, just from the values of the sums, than by throwing a coin and choosing Heads/Kloosterman, Tails/Salié. But I wonder if the shapes of the graph of partial sums (as drawn by the program…) could be used to extract more information to lead to a guess with better than even odds of being correct? Or if, at least, it could be used to shorten the length of time one would need to look at the slideshow before being sure of the answer from the distribution perspective?