Does a Kloosterman sum composed with a rational function exhibit square root cancellation?

Question

Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the quadratic character. It follows from Weil's proof of the Riemann hypothesis for curves (or more elementary considerations in the case of $SL(a,b)$) that for $a,b \neq 0$ that $|K(a,b)|, |SL(a,b)| \lesssim |F|^{1/2}$.

Informally, I'd like to know if we can prove square root cancellation in the composition of a Kloosterman (respectively, Salié) sum with a rational function. In particular:

Let $R(t)$ be a rational function in $t$. For $b \neq 0$ can one prove $$ |\sum_{t \in F}'KL(R(t),b)|\lesssim |F|$$ and/or $$|\sum_{t \in F}'SL(R(t),b) \lesssim |F|$$ where $\sum'$ denotes the exclusion of the poles of $R(t)$. Of course the implicit constant should depend on $R$ but be independent of the field?

I'm particularly interested in the special case when the numerator and denominator of $R$ have degree $2$, but would be very interested in the general case as well.

There's been a lot of work by Bombeiri, Deligne, Hooley, Katz, Sperber, and others on deducing multivariate exponential sum estimates from Deligne's theorems, but many of these theorems are hard to parse for someone who isn't an expert in algebraic geometry. I'd also be interested in the above sums twisted by a multiplicative character in $t$.

Answer 1

It's fine for Kloosterman sums, as long as R is non-constant. The simplest argument (formally) being to use Deligne's general form of the Riemann Hypothesis. Briefly, KL(R(t),b) is the trace function of the sheaf R^*Kl, where Kl is the Kloosterman sheaf of rank 2 defined by Deligne. Katz has shown that the geometric monodromy group of that is SL_2, so R^*Kl has the same geometric monodromy group (a priori it could become a finite-index subgroup, but SL_2 has no finite-index algebraic subgroup). Then the Grothendieck-Lefschetz trace formula and the Riemann Hypothesis apply to get what you want. This kind of things is sketched in Section 6 of the Polymath8 paper (see, e.g., Prop. 6.11), or in "A study in sums of products" by Fouvry-Kowalski-Michel, among other places, and the same argument applies with a twist by a multiplicative character