Trace functions, II: Examples

Continuing after my last post, this one will be a list of examples of trace functions modulo some prime number $p$. For each of the examples, I will give a bound for its conductor, which I recall is the main numerical invariant that allows us to measure the complexity of the trace function $K(n)$ (formally, the conductor is attached to the object $\mathcal{F}$ that gives rise to $K$, but we can define the conductor of a trace function to be the minimal conductor of such a $\mathcal{F}$.) These objects $\mathcal{F}$ will be called sheaves, since this is the language used in the paper(s) of Fouvry, Michel and myself, but one doesn’t need to know anything about sheaves to understand the examples.

I will start with a list of concrete functions which are trace functions, and then explain some of the basic operations one can perform on known trace functions to obtain new ones. All these examples will be (I hope) very natural, but it is usually a deep theorem that the functions come from sheaves.

Throughout, $p$ is a fixed prime number. Generically, $\psi$ denotes a non-trivial additive character modulo $p$, for instance
$\psi(x)=e^{2i\pi x/p},$
(which may also be viewed casually as an $\ell$-adic character), and $\chi$ denotes a multiplicative character modulo $p$ (non-trivial, unless specified otherwise.)

(1) Characters and mixed characters

Let $f$ and $g$ be non-zero rational functions in $\mathbf{F}_p(T)$. Let
$K(x)=\psi(f(x))\chi(g(x)),$
for $x$ which is not a pole of $f$, or a zero or pole of $g$, and $K(x)=0$ in that case. Then $K$ is a trace weight. The (or an) associated sheaf is of rank $1$, and its conductor is bounded by the sum of degrees of numerators and denominators of $f$ and $g$. However, the size of the conductor arises for different reasons for $f$ and $g$: for the “additive” component $f$, singularities are poles of $f$, and the contribution of each pole $x_0$ comes from the Swan conductor, which is bounded by the order of the pole at $x_0$; for the “multiplicative” component $g$, the singularities are zeros and poles of $g$, and each only contributes $1$ to the conductor: the Swan conductors for $K_g=\chi(g(x))$ are all zero.

For analytic applications, the main point is that, by fixing $f$ and $g$ over $\mathbf{Q}$, one obtains for each $p$ large enough (so that the reduction modulo $p$ makes sense), and each choice of characters $\psi$ and $\chi$, a trace weight associated to $f$ and $g$ which has conductor uniformly bounded (depending on $f$ and $g$ only). Thus any estimates valid for all primes with implied constants depending only on the conductor of the trace functions involved will become an interesting estimate concerning $f$ and $g$. This applies to the main theorem of my paper with Fouvry and Michel concerning orthogonality of Fourier coefficients of modular forms and trace functions…

These examples are the most classical, and are very useful. Even the simple case $g=1$ and $f(X)=X^{-1}$ is full of surprises.

(2) Fiber-counting functions

Another very useful example comes from a fixed non-constant rational function $f\in \mathbf{F}_p(T)$, which is viewed as defining a morphism
$f\,:\, \mathbf{P}^1\rightarrow \mathbf{P}^1.$
Consider then
$K(x)=|\{y\in \mathbf{P}^1\,\mid\, f(y)=x\}|.$
This is a trace weight, associated to the direct image sheaf
$\mathcal{F}=f_*\bar{\mathbf{Q}}_{\ell},$
which in representation theoretic terms is an induced representation from a finite-index subgroup, so that it remains relatively simple.
Here the rank $r$ of the sheaf is the degree $\deg(f)$ of $f$ as a morphism (i.e., the generic number of pre-images of a point $x$); the singularities are the finitely many $x$ in $\mathbf{P}^1$ such that the equation
$f(y)=x$
has fewer than $r$ solutions (in $\mathbf{P}^1(\bar{\mathbf{F}}_p)$) and, at least if $p>\deg(f)$, the Swan conductors vanish everywhere, so that the conductor is bounded in terms of the degrees of the numerator and denominator of $f$ only. In particular, if $f$ is defined over $\mathbf{Q}$, varying $p$ (large enough) will provide a family of trace functions modulo primes with uniformly bounded conductor, similar to the characters of the previous example with fixed rational functions as arguments.

The main reason this function is useful is that, for any other (arbitrary) function $\varphi$ on $\mathbf{P}^1(\mathbf{F}_p)$, we have tautologically
$\sum_{y}{\varphi(f(y))}=\sum_{x}{K(x)\varphi(x)}$
(in other words, it is maybe better to interpret $K$ as the image measure of the uniform measure on the finite set $\mathbf{P}^1(\mathbf{F}_p)$ under $f$, and this formula is the classical “integration” formula for an image measure…)

One also often takes the function
$\tilde{K}(x)=K(x)-1,$
where $1$ is the average of $K$ over $\mathbf{F}_p$. This is also a trace function (the sheaf corresponding to $K$ contains a trivial quotient, and this is the trace function of the kernel of the map to this trivial quotient). We now have
$\sum_{x}{\tilde{K}(x)\varphi(x)}=\sum_{y}{\varphi(f(y))}-\sum_{x}{\varphi(x)}.$

(3) Number of points on families of algebraic varieties

More generally, we can count points on one-parameter families of algebraic varieties of dimension $d\geq 1$. For instance, families of elliptic curves or of more general curves are quite common. To be concrete, one may have a polynomial $f\in \mathbf{F}_p[T,Y,Z]$, where $T$ is seen as the parameter, and consider the curves
$C_t\,:\, f(t,X,Y)=0.$
Usually, it is not so much the number of points as the correction term that is most interesting. For instance, if the curves are generically geometrically irreducible, and have a single point at infinity, the size of $C_t(\mathbf{F}_p)$ is (for all but finitely many $t$) of the form
$|C_t(\mathbf{F}_p)|=p-a(C_t),$
where $a_(C_t)$ satisfies the Weil bound
$|a(C_t)|\leq 2g(C_t)\sqrt{p},$
in terms of the genus of $C_t$. In fact, once one ensures that the family of curves is such that the genus of the curves is the same $g\geq 0$ (for all but finitely many $t$), the function
$K(t)=a(C_t)$
is a trace function on the corresponding dense open set of $\mathbf{A}^1$, for some sheaf which has rank $2g$. For the other values of $t$, the trace function of the corresppnding middle-extension sheaf might differ from the value $a(C_t)$ defined as above using the number of points, but since the number of those singularities is bounded by the conductor, one can usually (analytically at least) not worry too much about this. Similarly, in many cases the sheaf is tamely ramified everywhere (i.e., all Swan conductors vanish), and so the conductor is well-controlled.

In contrast with the first two examples, the construction of a sheaf with this trace function is not elementary: it is an example of the so-called “higher direct image sheaves” (with compact support). Since, for every “good” $t$, the Riemann Hypothesis for curves shows that
$a_p(C_t)=\sqrt{p}(\theta_{1,t}+\cdots+\theta_{2g,t}),$
where the $\theta_{i,t}$ are complex numbers of modulus $1$, we can interpret the existence of this sheaf as saying that the algebraic variation of the “eigenvalues” $\theta_{i,t}$ is itself controlled by an algebraic object. This is one of the main insights that algebraic geometry (and étale cohomology in particular) brings to analytic number theory.

The family of elliptic curves
$x+x^{-1}+y+y^{-1}+t=0$
in my bijective challenge is of this type.

(4) Families of Kloosterman sums

One of the great examples, for analytic number theory, is given by families of Kloosterman sums: for an integer $m\geq 1$, and a non-zero $a\in\mathbf{F}_p$, we let
$Kl_m(a)=\frac{(-1)^{m-1}}{p^{(m-1)/2}}\sum_{x_1\cdots x_m=a}e\Bigl(\frac{x_1+\cdots +x_m}{p}\Bigr).$
The Weil bound for $m=2$, and the even deeper work of Deligne for larger $m$, prove that
$|Kl_m(a)|\leq m$
for all $a$ invertible modulo $p$. Further work, relying once more on the powerful formalism of étale sheaves and higher direct images in particular, shows that the function
$K(a)=Kl_m(a),$
is (the restriction to invertible $a$ of) a trace function for an irreducible sheaf, with conductor bounded in terms of $m$ only.

(5) The Fourier transform

If we have a function $K(x)$ modulo $p$, we define its Fourier transform by
$\hat{K}(t)=\frac{1}{\sqrt{p}}\sum_{x\in \mathbf{F}_p}{K(x)e\Bigl(\frac{xt}{p}\Bigr)}$
for $t\in\mathbf{F}_p$ (the normalization here is convenient, as I will explain). It is now a very deep fact that, if $\latex K$ comes from a sheaf, then so does $-\hat{K}$ (the minus sign is natural, but this has to do with rather deep algebraic geometry…) More precisely, one has to be careful because of the fact that the Fourier transform of an additive character (as a function) is a multiple of a delta function. The latter does fit nicely in the framework of étale sheaves, but not as a middle-extension sheaf or Galois representation (because it is zero on a dense open set, so it would have to be zero to be a middle-extension sheaf or to come from a Galois representation). There is a geometric solution to this issue, but it involves speaking of perverse sheaves and related machinery, which we have barely started to understand: the Fourier transform works perfectly well at the level of perverse sheaves, and one can use their trace functions just as well as those of Galois representations. Since, in our current applications, we can always deal separately with additive characters (or delta functions), we have avoided having to deal with perverse sheaves (up to now…)

The existence of the $\ell$-adic Fourier transform of sheaves was first proved by Deligne, but the theory of the sheaf-theoretic Fourier transform was largely built by Laumon (with further contributions, in particular, from Brylinski and Katz). To illustrate how powerful it is, consider
$K(x)=e\Bigl(\frac{x^{-1}}{p}\Bigr),$
a relatively simple case of Example (1). We then have
$\hat{K}(x)=Kl_2(x),$
so that the existence of the Fourier transform at the level of sheaves implies the existence of the Kloosterman sheaf parameterizing classical Kloosterman sums as in the previous example.

Other examples that arise from our previous examples are many families of exponential sums, for instance
$K(t)=\frac{1}{\sqrt{p}}\sum_{x\in\mathbf{F}_p}{\psi(f(x)+tx)\chi(g(x))},$
(arising from Example (1); one must assume either that $f(x)$ is not a polynomial of degree $\leq 1$ or that $\chi$ is non-trivial to have a well-defined sheaf), or
$K(t)=\frac{1}{\sqrt{p}}\sum_{x}{e\Bigl(\frac{tf(x)}{p}\Bigr)},$
for $t\not=0$ with $K(0)$ equal to the number of poles of $f$ (the sum over $x$ is over values where the rational function $f$ is defined), that arises from Example (2) (applied with the function $\tilde{K}$).

This operation of Fourier transform has one last crucial feature for applications to the analysis of trace functions: the conductor of $\hat{K}$ is bounded in terms of that of $K$ only. This is something we prove in our paper using Laumon’s analysis of the singularities of the Fourier transform, and in fact we show that if the conductor of $K$ is at most $M\geq 1$, then the conductor of $\hat{K}$ is at most $10M^2$. Hence the examples above, if the rational functions $f$ (and/or $g$) are fixed in $\mathbf{Q}(T)$ and then reduced modulo various primes, always have conductor bounded uniformly for all $p$.

(6) Change of variable

Given a non-constant rational function $f\in\mathbf{F}_p(T)$ seen as a morphism
$\mathbf{P}^1\rightarrow \mathbf{P}^1,$
and a trace function $K(x)$, one can form the function
$f^*K(x)=K(f(x)).$
This is again, essentially, a trace function: as in Example (3), one may have to tweak the values of $f^*K$ at some singularities (because pull-back of middle-extension sheaves do not always remain so), but this is fairly easily controlled. Moreover, one can also control the conductor of $f^*K$ in terms of that of $K$, taking into account the degree of latex f\$. A specially simple case of great importance is when $f$ is an homography
$f(x)=\frac{ax+b}{cx+d},\quad\quad\quad ad-bc\not=0,$
(an automorphism of $\mathbf{P}^1$) in which case no tweaking is necessary to defined $f^*K$, and the conductor is the same as that of $K$ (which certainly seems natural!)

We can now compose these various operations. One construction is the following (a finite-field Bessel transform): start with $K$, apply the Fourier transform, change the variable $t$ to $t^{-1}$, apply again the Fourier transform. If we call $\check{K}$ the resulting function, the examples above show that if $K$ is a trace function with conductor $\leq M$, then $\check{K}$ will also be one, and its conductor will be bounded solely in terms of $M$ (in fact, it will be $\leq 100M^4$ by the bound discussed in Example (5)).

Trailer! In the next post in this series, I will discuss the Riemann Hypothesis for trace functions and its applications. But probably before I will discuss the more recent works of Fouvry, Michel and myself, since we now have three further papers in our series — two small, and one big.