## The support of Kloosterman paths

Will Sawin and I just put up on arXiv a preprint that is the natural follow-up to our paper on those most alluring of shapes, the Kloosterman paths.

As the title indicates, we are looking this time at the support of the limiting random Fourier series that arose in that first paper, namely
$K(t)=t\mathrm{ST}_0+\sum_{h\not=0}\mathrm{ST}_h\frac{e^{2i\pi ht }-1}{2i\pi h},$
where $(\mathrm{ST}_h)_{h\in\mathbf{Z}}$ is a sequence of independent Sato-Tate-distributed random variables. In a strict sense, this should be a very short paper, since the computation of the support is easily achieved using some basic probability and elementary properties of Fourier series: it is the set of continuous functions $f\colon [0,1]\to \mathbf{C}$ such that (1) the value of $f$ at $t=1$ is real and belongs to $[-2,2]$; (2) the function $g(t)=f(t)-tf(1)$ has purely imaginary Fourier coefficients $\hat{g}(h)$ for $h\not=0$; (3) we have $|\hat{g}(h)|\leq 1/(\pi |h|)$ for all $h\not=0$.

So why is the paper 26 pages long? The reason is that this support (call it $\mathcal{S}$) is a rather interesting set of functions, and we spend the rest of the paper exploring some of its properties. Most importantly, the support is not all functions, so we can play the game of picking our favorite continuous function on $[0,1]$ (say $f_0$) and ask whether or not $f_0$ belongs to $\mathcal{S}$.

For instance:

1. Fixing a prime $p_0$, and $a_0$, $b_0$ invertible modulo $p_0$, does the Kloosterman path $K_{p_0}(a_0,b_0)$ itself belong to the support? Simple computations show that it depends on $(p_0,a_0,b_0)$! For instance, the path for the Kloosterman sum $\mathrm{Kl}_2(8,1;9)$, shown below, does not belong to the support. (As we observe, it looks like a Shadok, whose mathematical abilities are well-known — sorry, the last link is only in French ; I suggest to every French-aware reader to watch the corresponding episode, since the voice of C. Piéplu achieves the seemingly impossible in making this hilarious text even funnier…)
2. On the other hand, the path giving the graph of the Takagi function $T$ (namely $f(t)=t+iT(t)$) belongs to the support.
3. But maybe the most interesting problem from a mathematical point of view is one of pure analysis: when we see a Kloosterman path (such as the one above), we only see its image as a function from $[0,1]$ to $\mathbf{C}$, independently of the parameterization of the path. So we can take any shape in the plane that can be represented as the image of a function $f$ satisfying the conditions (1) and (2) above, and ask: is there a reparameterization of $f$ that belongs to the suppport? For instance, for the Kloosterman paths themselves (as in (1) above), it is not difficult to find one: instead of following each of the $p_0-1$ segments making the Kloosterman path in time $1/(p_0-1)$, one can insert a “pause” of length $1/(2p_0)$ at the beginning and end of the path, and then divide equally the remaining time for the $p_0-1$ segments. (The fact that this re-parameterized path, whose image is still the same Kloosterman path, belongs to the support $\mathcal{S}$ is then an elementary consequence of the Weil bound for Kloosterman sums).

4. In general, the question is whether a given $f$ has a reparameterization with Fourier coefficients (rather, those of $t\mapsto f(t)-tf(1)$) are all smaller than $1/(\pi |h|)$. This is an intriguing problem, and looking into it brought us into contact with some very nice classical questions in Fourier analysis, that I discuss in this later post. We only succeeded in proving the existence of a suitable reparameterization for real-valued functions, for reasons explained in the aforementioned later post, and it is an interesting analysis problem whether the result holds for all functions. A positive answer would in particular settle another natural question that we haven’t been able to handle yet: is there a space-filling curve in the support of the Kloosterman paths?

All this is great analytic fun. But there are nice arithmetic consequences of our result. By the definition of the support, we know at least that any $f\in\mathcal{S}$ has the property that, with positive probability, the actual path of the partial sums of the Kloosterman sums will come as close as we want (uniformly on $[0,1]$) to $f$, and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime $p$, there is a positive proportion of $(a,b)\in \mathbf{F}_p^{\times}\times\mathbf{F}_p^{\times}$ such that all partial sums
$\frac{1}{\sqrt{p}}\sum_{1\leq x\leq j}\exp(2i\pi(ax+b\bar{x})/p),$
for $1\leq j\leq p-1$, have modulus $<\varepsilon$.
In other words, there is a non-zero probability that all the normalized partial sums of the Kloosterman sums are very small. (It is interesting to note that this is emphetically not true for character sums… the point is that their Fourier expansion involves multiplicative coefficients, so they cannot become smaller than $1/(\pi |h|)$.)

## Update on a bijective challenge

A long time ago, I presented in a post a fun property of the family of curves given by the equations
$x+1/x+y+1/y=t,$
where $t$ is the parameter: if we consider the number (say $N_p(t)$) of solutions over a finite field with $p\geq 3$ elements, then we have
$N_p(t)=N_p(16/t),$
provided $t$ is not in the set $\{0, 4, -4\}$. This was a fact that Fouvry, Michel and myself found out when working on our paper on algebraic twists of modular forms.

At the time, I knew one proof, based on computations with Magma: the curves above “are” elliptic curves, and Magma found an isogeny between the curves with parameters $t$ and $16/t$, which implies that they have the same number of points by elementary properties of elliptic curves over finite fields.

By now, however, I am aware of three other proofs:

• The most elementary, which motivates this post, was just recently put on arXiv by T. Budzinski and G. Lucchini Arteche; it is based on the methods of Chevalley’s proof of Warning’s theorem: it computes $N_p(t)$ modulo $p$, proving the desired identity modulo $p$, and then concludes using upper bounds for the number of solutions, and for its parity, to show that this is sufficient to have the equality as integers.  Interestingly, this proof was found with the help of high-school students participating in the Tournoi Français des Jeunes Mathématiciennes et Mathématiciens. This is a French mathematical contest for high-school students, created by D. Zmiaikou in 2009, which is devised to look much more like a “real” research experience than the Olympiads: groups of students work together with mentors on quite “open-ended” questions, where sometimes the answer is not clear (see for instance the 2016 problems here).
• A proof using modular functions was found by D. Zywina, who sent it to me shortly after the first post (I have to look for it in my archives…)
• Maybe the most elegant argument comes by applying a more general result of Deligne and Flicker on local systems of rank $2$ on the projective line minus four points with unipotent tame ramification at the four missing points (the first cohomology of the curves provide such a local system, the missing points being $0, 4, -4, \infty$). Deligne and Flicker prove (Section 7 of their paper, esp. Prop. 7.6), using a very cute game with products of matrices and invariant theory, that if $\mathcal{V}$ is such a local system and $\sigma$ is any automorphism of the projective line that permutes the four points, then $\sigma^*\mathcal{V}$ is isomorphic to $\mathcal{V}$.

Not too bad a track record for such a simple-looking question… Whether there is a bijective proof remains open, however!

## L-functions database!

People studying automorphic forms, automorphic representations, number fields, diophantine equations, function fields, algebraic curves, equidistribution and many other arithmetic objects (j’en passe, et des meilleurs), often end up with some “L-function” to deal with — indeed, probably equally often, with a whole family of them, sometimes not so well-behaved… These objects are fascinating, mystifying, exhilarating, random and possibly spooky. Where they really come from is still a mystery, even with buzzwords aplenty ringing around our ears. But one remarkable thing was already known to Euler and to Riemann: one can compute with L-functions. One impressive research project has been building, for quite a few years, a very sophisticated website presenting enormous amounts of data about L-functions of many kinds. The L-functions Database is now out of its beta status: go see it, and have a look at the list of editors to see who should be thanked for this amazing work!

## New versions, new bugs

Ph. Michel found the first bug in the new version of Kloostermania: when entering a modulus for a sum from the menu, there was no check that it is prime (or adjustment to make it so). This is now corrected in version 1.01, to be found in the usual place.

As another bonus feature, a double tap on the screen will cycle between the types of sums presented: Kloosterman sum to Birch sum to both to Kloosterman…

## Kloostermania turns 1.0

Since the Pocket Kloostermania has remained unchanged for almost five years, an update is probably welcome! This is now available, in time to honor N. Katz on the occasion of this 71st birthday (which I will help celebrate in China next week).

Important changes in the new version:

(1) It was recompiled — hence a new modern look instead of a style reminiscent of the dark ages –, and the resulting binary should work on any Android system with version 4.0.3 or later;

(2) To keep up with recent progress, the program now displays Kloosterman sums and/or Birch sums, instead of Kloosterman and/or Salié sums. On the other hand, the moduli used are now restricted to primes, to simplify things a bit, and only one parameter is used: the sums displayed are
$\sum_{x\in\mathbf{F}_p^{\times}}e\Bigl(\frac{ax+\bar{x}}{p}\Bigr)$
and
$\sum_{x\in\mathbf{F}_p}e\Bigl(\frac{ax+x^3}{p}\Bigr)$

(3) In addition to being able to change the modulus by swiping horizontally, a vertical swipe on the screen scrolls among the values of the parameter $a$. As was the case in the previous version (and even more because phones are faster now), the scrolling is usually too fast for a single step, so tapping once close to one of the edges of the window displaying the sum will perform just one step of the corresponding move (e.g., tapping close to the right of the screen goes to the next prime modulus). The value of the sum and its parameters and then displayed quickly.

(4) The plots are presented in orthonormal configurations, to give a more faithful representation of the paths in the plane;

(5) In the “About” dialog, a single click will display (if a PDF viewer is installed) the paper of W. Sawin and myself that explains the limiting distribution of Kloosterman (and Birch) paths…

(6) It is possible to save (or rather to “share” in the usual Android way) a picture of the sums currently displayed, as a PNG file.

(6) And the launcher icon is better.

The installation file is available right now on the updated Kloostermania page!