4
$\begingroup$

Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define $$ K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)), $$ where $e_p(t):=e^{2\pi i \frac{t}{p}}$.

It is well-known that if $p\nmid d$, then $|K|$ is bounded by $1$ (up to some constant depending on $d$).

Now consider a sum of products for fixed $t\neq 0$ and $y\neq y'$: $$ I=\sum_{x\in\mathbb{F}_p}K(x,y)\overline{K(x-t,y+t)}\overline{K(x,y')}K(x-t,y'+t). $$ My question is: for what $f\in\mathbb{F}_p[X]$ can we bound $|I|$ by $\sqrt{p}$ (up to a multiplicative constant, of course), or perhaps a little worse: $p^{\delta}$ ($\delta<1$). When $f$ is quadratic, then $K$ is a Gauss sum that can be evaluated explicitly, and I can calculate $I$ and answer my question affirmatively. I don't know what happens for general $f$.

$\endgroup$

2 Answers 2

6
$\begingroup$

Here is a different approach than Denis Chaperon de Lauzières's.

Opening everything and using orthogonality of characters to remove the $x$ variable, we see that your sum is $1/p$ times $$\sum_{z_1,z_2,z_3,z_4 \in \mathbb F_p, z_1-z_2-z_3+z_4 =0 } e_p \left( y f(z_1) + t z_2 - (y+t) f(z_2) - y' f(z_3) -t z_4 + (y'+t) f(z_4) \right)$$

This is an exponential sum of a $3$-variable polynomial. Deligne's theorem (Theorem 8.4 of Weil I) shows that this is $O(p^{3/2})$ (with explicit constant $(d-1)^{3}$) as long as $d$ is prime to $p$ and the leading term of the polynomial inside the $e_p$ defines a smooth hypersurface.

The leading term looks like $$y z_1^d - (y+t) z_2^d - y' z_3^d + (y'+ t) z_4 ^d $$

The intersection of this with $z_1 - z_2 - z_3 + z_4 =0$ is smooth unless $$ y^{-1/(d-1)} - (y+t)^{-1/(d-1)} - y' ^{-1/(d-1)} + (y'+t)^{1/d-1}=0$$.

So you get square-root cancellation outside this set for arbitrary $f$.

$\endgroup$
1
  • $\begingroup$ Thank you for the detailed calculation. For each $y$, there are only $O(1)$ $y'$ that make the hypersurface non-smooth, which is acceptable in applications. $\endgroup$
    – Tony B
    Jun 11, 2017 at 4:16
5
$\begingroup$

Estimates for such sums can often be deduced from Deligne's version of the Riemann Hypothesis using computations of the relevant monodromy groups (typically by Katz, using sheaf-theoretic Fourier-transform arguments) and algebraic-group results (what Katz calls the Goursat-Kolchin-Ribet argument). For analytic number theory purposes, there's a survey called "A study in sums of products" (Fouvry, Kowalski, Michel) that you can find online, with examples and pointers to the literature.

$\endgroup$
2
  • $\begingroup$ Thank you for the reference. I notice that Corollary 3.3 in that paper answers my question when $K$ is a Kloosterman sum. I'm not quite sure if the paper also considers the Weil's sum case. $\endgroup$
    – Tony B
    Jun 10, 2017 at 19:58
  • $\begingroup$ In this case, the relevant monodromy group might not be so easy to compute. I don't think Katz said much on this kind of sheaves. His student Ondrej Such did in his thesis, but I didn't find in it a clean statement of the kind you need for Goursat-Kolchin-Ribet. semanticscholar.org/paper/… $\endgroup$
    – Will Sawin
    Jun 10, 2017 at 20:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.