# Three new papers

In this post, I will just describe briefly three papers (one long, two short) that É. Fouvry, Ph. Michel and myself have finished in recent weeks and days concerning the properties of trace functions. The last one should be on arXiv tomorrow, the others are there already. I will probably say more about some (or all) of these papers later, but here are quick summaries of what we do…

This is a fairly short note, where we use the quasi-orthogonality of trace functions (in the geometrically irreducible case), which encapsulates Deligne’s general form of the Riemann Hypothesis over finite fields, in order to derive upper-bounds for the number of such functions with bounded conductor over a given finite field. As it turns out, the same quasi-orthogonality implies that we do something more geometrically interesting: for “small enough” conductor, the trace function essentially determines the sheaf, and so we are counting sheaves.

In spirit, this is therefore close to many counting problems of number theory: we have a countable set, a measure of complexity which allows us to write it as an increasing union of finite sets, and we want to know how many elements there are in these finite subsets.

A difference with many more classical problems, however, is that it seems rather difficult to get asymptotics when counting trace functions. If we use a Langlands correspondance, we are trying to count automorphic representations on $\mathrm{GL}_n(\mathbf{F}_p[T])$ with some bounds on ramification. We realized only rather late the existence of very striking conjectures and results of Drinfeld and Deligne (among others; see this excellent account of Deligne’s work by Esnault and Kerz) for the precise counting in the vertical direction (fixing a base field, and extending it), which should — under suitable conditions — take a form very similar to a Lefschetz Trace Formula. Our bounds do not really contribute to this question, since they are (probably far off) upper-bounds only, but they are completely explicit and they work in the “horizontal” direction (bounding the conductor and letting $p$ go to infinity.)

As to the spherical codes of the title, they arise because the quasi-orthogonality shows that, as vectors in the $2p$-dimensional real vector space of complex-valued functions on $\mathbf{F}_p$, the (normalized) trace functions with conductor $\leq M$ have a strong angular-separation property, and subsets of unit spheres with this property are precisely called “spherical codes”. The question of giving upper-bounds for the cardinality of spherical codes with given angular separation is quite important, but interestingly we did not find the range given by the Riemann Hypothesis in the literature (this has the effect of making the cardinality grow polynomially as a function of $p$ for a fixed bound on the conductor; a polynomial growth is the right answer for our problem, though finding the right exponent is a rather delicate question). We tweaked the ideas of Kabatjanski and Levenshtein (who have the best-known results in general) for our purpose, which involves some fun estimates depending on the location of the first zero of the Airy function…

This is again a fairly short paper, which does pretty much what the title suggests: for a trace function $K$ over $\mathbf{F}_p$, and an integer $d\geq 1$, we find an estimate for the $d$-th Gowers norm $\|K\|_{d}$ of $K$ (see Section 3 of this part of T. Tao’s notes on higher-order Fourier analysis for an introduction to these norms). This takes the form
$\|K\|_{d}^{2^d}\ll p^{-1},$
where the implied constant depends only (completely explicitly) on the conductor of $K$ and on $d$, except (this is the inverse part in the title) when the sheaf $\mathcal{F}$ which gives rise to $K$ contains at least one Jordan-Hölder factor with trace function of the type
$e\Bigl(\frac{P(x)}{p}\Bigr),$
for some polynomial $P$ of degree $\leq d-1$. These functions are the natural obstructions to having small Gowers norms (as already emphasized by Gowers), but one doesn’t usually get to have such strong structural statements as those we get: if $K$ is geometrically irreducible, then the only possibility is that it is exactly proportional to a function of the type above (for all $x$).

None of the three of us can be said to be a great expert on the Gowers norms (which have been studied much more deeply by others, most spectacularly by Gowers and recently by Green, Tao and Ziegler), and this note is basically our attempt at seeing if the (fairly algebraic) definition could be studied using the sheaf formalism and the Riemann Hypothesis. But the final estimate is interesting in that, as far as the dependency on $p$ is concerned, it is the same as one would get for a “random” function (in the model where we consider a function $\varphi$ modulo $p$ such that the $\varphi(x)$ are independent uniformly bounded random variables with mean zero; we found this statement in the book of Tao and Vu), and it seems that no “deterministic” examples of such functions had been written down before. From our result, one can see for instance that
$\|x\mapsto \chi(f(x))\|_d^{2^d}\ll p^{-1},$
for any fixed non-constant polynomial $f\in \mathbf{Z}[T]$, $\chi$ being the Legendre character modulo $p$, with the implied constant depending only on $\deg(f)$ and $d$ (again, completely explicitly).

The longest and deepest of our three papers continues the study of orthogonality of trace functions against other natural arithmetic sequences. After dealing with Fourier coefficients of modular forms in the first paper, we consider sums over primes, and sums against the Möbius function. Precisely, let $K$ be a trace function modulo $p$. Say that $K$ is $p$-exceptional if it is proportional to
$K_{\chi,a}(x)=\chi(x)e(ax/p),$
for some Dirichlet character $\chi$ modulo $p$ and some $a\in\mathbf{F}_p$ (allowing trivial $\chi$ and/or $a=0$.) Then, if $K$ is not $p$-exceptional we have
$\sum_{n\leq X}\Lambda(n)K(n)\ll X\Bigl(1+\frac{p}{X}\Bigr)^{1/12}p^{-1/48+\varepsilon}$
for any $\varepsilon>0$, where the implied constant depends only on the conductor of $K$ and on $\varepsilon$. The “critical” case is when $X=p$ or is a bit smaller, in which case we therefore get cancellation with a power saving. It is well-known that one expects such a bound for a $p$-exceptional $K$ also, but that this is essentially equivalent to proving the existence of a zero-free strip for some Dirichlet $L$-function, so that the restriction is natural in the current state of knowledge.

Similarly, we have
$\sum_{n\leq X}\mu(n)K(n)\ll X\Bigl(1+\frac{p}{X}\Bigr)^{1/12}p^{-1/48+\varepsilon}$
with the same conditions.

These estimates are rather sweeping: we can take any of the examples of trace functions explained in the previous post (making sure they are not exceptional, but for instance any irreducible sheaf of rank at least $2$ is not exceptional, as is any rank $1$ sheaf with a singularity not at $0$ or $\infty$…). Although some specializations to specific trace functions had been already studied (sometimes with stronger exponents), we find the generality to be a really remarkable example of the power of the structural features coming from Deligne’s work. and from the formalism of algebraic geometry, which we use again extensively. Indeed, we need not only all the work of the previous paper on twists of Fourier coefficients of modular forms (applied to Eisenstein series), but we also had to establish some additional sheaf-theoretic properties.

To give an example, we get immediately that if $\chi$ is a Dirichlet character of order $h\geq 2$, and $f\in \mathbf{Z}[T]$ is a polynomial which is not proportional to an $h$-th power times a monomial (e.g., if $f$ is squarefree) we have
$\sum_{n\leq X}\Lambda(n)\chi(f(n))\ll X\Bigl(1+\frac{p}{X}\Bigr)^{1/12}p^{-1/48+\varepsilon}$
where the implied constant depends only on $\deg(f)$ and on $\varepsilon$. As far as we know, the only case previously treated (going back to Karatsuba) is when $f(x)=aX+b$, with $b\not=0$, is linear…

Among a number of applications, which can be found in the paper (and before we find others…), the following is also fairly nice: given $f\in\mathbf{Z}[T]$ squarefree and non-constant, we have
$\sum_{0\leq a
where $E(X,q,a)$ denotes in general the error term in the prime number theorem in arithmetic progressions
$\sum_{p\leq X,\ p\equiv a\bmod q}1=\frac{\pi(X)}{\varphi(q)}+E(X;q,a)$
and the implied constant depends only on $\deg(f)$ and on $\varepsilon$. In fact, with a whiff of extra formalism, we can replace the sum over residue classes of the form $f(a)$ taken with the multiplicity of representation to the corresponding sum over the the residues of this form, without multiplicity.