Estimates of Kloosterman sums

This is motivated from a talk presented by prof. Ngo Bao Chau that I attended recently. The subject of the talk was to give an estimate of the so-called Kloosterman sum by transferring it to the language of $l$-adic cohomology. Follow Katz, let me spend some momemt to recall the motivating problem: given a prime $p$ and an integer $a$ s.t. $(a,p)=1$, the Kloosterman sum is defined as the complex number

$$\mathrm{Kl}(a,p) = \sum_{(x,y) \in \mathbb{F}_p: xy = a} \operatorname{exp}\left(\frac{2\pi i}{p}(x+y) \right).$$ By an elementary argument, one can see that this sum is a real number and in the early time when Kloosterman studied the Hardy-Littlewood circle method, he wanted to bound this sum by a function of $p$. 

Some motivations 

Theorem 1 (Kloosterman 1926). For any $\epsilon > 0$, we have $\left |\mathrm{Kl}(a,p) \right| < Cp^{3/4+\epsilon}$.  

Kloosterman's proof was quite elementary, however, the bound can be sharpen much more as follows.

Theorem 2 (Weil). We have $\left |\mathrm{Kl}(a,p) \right| \leq 2\sqrt{p}$.

This estimate is a consequence of Weil's proof of the "early Riemann hypothesis". The analytic version of Kloosterman sums is

$$\int_{-\infty}^{\infty}e^{i(ax+x^{-1})}dx$$ which is clearly not convergent, but we can approximate it by $\sqrt{a}K(\sqrt{a})$ where $K$ is the Bessel function. More generally, one can consider the Kloosterman sum

$$\mathrm{Kl}(a,p) = \sum_{xy=a \in \mathbb{F}_p} \psi(x+y) = \sum_{x \in \mathbb{F}_p}\psi(ax+x^{-1})$$ for any character $\psi:\mathbb{F}_p \longrightarrow \mathbb{C}^{\times}$, i.e. $\psi(x+y)=\psi(x)\psi(y)$. Here we can also prove that $\left| \mathrm{Kl}(a,p) \right| \leq 2 \sqrt{p}$ but even more, we can prove that

$$\mathrm{Kl}(a,p) = \alpha + \overline{\alpha}$$ where $\alpha$ is a complex number with $\left| \alpha \right| = \sqrt{p}$. This remains true if we replace $p$ by some of its power. This is where algebraic geometry enters the play. The first task is to transfer functions to sheaves. At the level of sheaves, we have more operations to manipulate (at least functions do not have something like duality).  But before one can see why we have to translate everything to cohomology language, one needs to have some clues about Grothendieck's formalism of six operations in $l$-adic cohomology.

$l$-adic cohomology

Let's fix a finite field $k=\mathbb{F}_{q}$ (where $q = p^n$ and $p$ prime) and $X/k$ be a variety. Given an integer $n$ invertible on $k$, then we can define that derived category $D^b_c(X,\mathbb{Z}/n)$ of chain complexes (modulo quasi-isomorphisms) of etale sheaves having cohomology sheaves are constructible. If $l \neq p$ is another prime, we define

$$D^b_c(X) = D^b_c(X, \overline{\mathbb{Q}}_l) = \left( \underset{\longleftarrow}{\lim} \ D^b_c(X,\mathbb{Z}/l^n\mathbb{Z}) \right) \otimes_{\mathbb{Z}_l} \overline{\mathbb{Q}}_l.$$ This definition is subtle and technical so one might follow Bhatt and Scholze's instruction to pretend that $D^b_c(X,\overline{\mathbb{Q}}_l)$ is some full subcategory of a derived category $D^b(X,\overline{\mathbb{Q}}_l)$. This is in fact does not cause any harm because almost every result for $D^b_c(X)$ is already true at the level $D^b_c(X,\mathbb{Z}/n\mathbb{Z})$. As far as I understand, the dissatisfaction with this limit-taking step is one of the reasons why Scholze introduced the pro-etale site.

Denote by $\mathfrak{TR}$ to be $2$-category of essentially small triangulated categories, then the family 

$$D^b_c: \mathrm{Var}/k \longrightarrow \mathfrak{TR} \  \ X \longmapsto D^b_c(X)$$ defines a $2$-functor admitting a formalism of six operations $(f^*,f_*,f_!,f^!,\otimes,\underline{\mathrm{Hom}})$, e.g. proper + smooth base change theorems, purity, Poincare duality,...

Objects of $D^b_c(X)$ are called $\mathbb{Q}_l$-sheaves or $l$-adic sheaves. The tensor product admits a unit denoted $\mathbb{Q}_{l,X}$ corresponding to the "constant" $l$-adic sheaf. For a $l$-adic sheaf $\mathcal{F}$, we define the $i$-th $l$-adic cohomology by setting

$$H^i(X \otimes_k \overline{k},\mathcal{F} \otimes_k \overline{k}) = \mathrm{Hom}_{D^b_c(X)}(\mathbb{Q}_{l,X},p_*\mathcal{F}[n]).$$ if $p: X \longrightarrow \mathrm{Spec}(k)$ is the structural morphism. Similarly, 

$$H^i_c(X \otimes_k \overline{k},\mathcal{F} \otimes_k \overline{k}) = \mathrm{Hom}_{D^b_c(X)}(\mathbb{Q}_{l,X},p_!\mathcal{F}[n]).$$ There is a subcategory of this category called smooth $l$-adic sheaves. Instead of treating (smooth) $l$-adic sheaf as complexes, we follow a shorter path:

Theorem 3. Let $X/k$ be an algebraic variety and $\overline{x} \longrightarrow X$ be a geometric point, then there is an equivalent of categories

$$\left \{\text{etale} \ \overline{\mathbb{Q}}_l-\text{sheaves} \right \} \overset{\sim}{\longrightarrow} \left \{\text{continuous rep. of} \ \pi_1(X,\overline{x}) \ \text{of} \ \overline{\mathbb{Q}}_l-\text{vector spaces} \right \}.$$

and moreover, smooth $l$-adic sheaves correspond to those representations which are of finite dimension. The equivalence is given by sending each etale $\overline{\mathbb{Q}}_l$ to its fiber over $\overline{x}$. 

About Frobenii 

During the study of this subject, I found out that the definition of the Frobenius morphism is ambiguous, precisely, there are several definitions of Frobenii, and the question is: which one is the right one that is used in our calculations and how are they related to others? I'll discuss few approaches to this definition, the explicit one and the abstract one. We still fix $k = \mathbb{F}_q$ and $k_n= \mathbb{F}_{q^n}$, the unique finite extension of degree $n$ of $k$.

Explicit definition

Although there are some different notions, they all arise from a single one, namely, the absolute Frobenius.

Definition. Let $A/k$ be an algebra, the Frobenius endomorphism $\mathrm{Frob}:A \longrightarrow A$ is simply the ring homomorphism $a \longmapsto a^p$. This construction is carried to schemes as it should be: if $X/k$ is a scheme, then the absolute Frobenius endomorphism $\mathrm{Frob}: X \longrightarrow X$ is a homeomorphism at the level of underlying topological spaces but on the structure sheaf is $f \longmapsto f^p$. Alternatively, it is defined locally by the Frobeninus endomorphism of affine pieces.

Caution. The Frobenius endomorphism is not an isomorphism in general. 

Lemma. The Frobeinus endomorphism $\mathrm{Frob}: X \longrightarrow X$ is finite of degree $q^{\dim(X)}$.

Proof. I strongly recommend you to prove this result with $X = \mathrm{Spec}(k[x_1,...,x_n])$ and move to the general case. Otherwise you can look at Milne's note.

Lemma 4. If $f: X \longrightarrow Y$ is a morphism of $k$-schemes, then $\mathrm{Frob}_Y \circ f = f  \circ \mathrm{Frob}_f$. In other words, the Frobenius construction is natural. 

Proof. Obvious. 

Much much more stronger is the following.

Theorem 5. If $f: U \longrightarrow X$ is an etale morphism of $k$-varieties, then the diagarm $$\require{AMScd}
\begin{CD}
 U @>{\mathrm{Frob}_U}>> U \\
@V f VV @V f VV \\
 X @>{\mathrm{Frob}_X}>> X.
\end{CD}$$

is cartesian.

Proof. By the previous lemma, there exists a canonical morphism, which is called the relative Frobenius morphism $\mathrm{Frob}_{X/U}: U \longrightarrow X \times_X U$. Note that since $f$ is etale, its base change, the projection onto the first factor $pr_X: X \times_X U \longrightarrow X$ is also etale. But $pr_X \circ \mathrm{Frob}_{X/U} = f$ so that $\mathrm{Frob}_{X/U}$ is etale. The absolute Frobenii are universally bijective (as noted in the definition), this forces $\mathrm{Frob}_{X/U}$ to be universally bijective. A morphism which is universally bijective and etale must be an isomorphism due to StackProject.

We can consider others Frobenii

• The relative Frobenius $\mathrm{Frob}_r = \mathrm{Frob}_X \times \mathrm{id}_{\overline{k}}: X \otimes_k \overline{k} \longrightarrow X \otimes_k \overline{k}$. This one is a special case of the one in the proof above.
  
• The arithmetic Frobenius $\mathrm{Frob}_a = \mathrm{id}_X \times \mathrm{Frob}_{\overline{k}}:X \otimes_k \overline{k} \longrightarrow X \otimes_k \overline{k}$.
• The geometric Frobenius $\mathrm{Frob}_g = \mathrm{id}_X \times \mathrm{Frob}_{\overline{k}}^{-1}:X \otimes_k \overline{k} \longrightarrow X \otimes_k \overline{k}$.
  

The relative and arithmetic are automorphisms while the geometric and the absolute are not.

Lemma 6. Given a variety $X/k$, then we have $X(k_r)  = \overline{X}^{\mathrm{Frob}_r^n}$ where the relative Frobenius acts on $\overline{X}$ on the first factor. In other words, the set of $k_n$-points of $X$ is the set of closed points of $\overline{X}$ which is fixed under the $r$-iteration of the Frobenius.

Proof. Check on affine pieces.  

The next point is to formulate the Grothendieck trace formula, which (I think people may not drop this point at the first reading) is our main tool of computation. We have to find a natural way to define an endormophism, denoted $\mathrm{Frob}^*$

$$\mathrm{Frob}^*: H^i_c(X \otimes_k \overline{k}, \mathcal{F} \otimes_k \overline{k}) \longrightarrow H^i_c(X \otimes_k \overline{k}, \mathcal{F} \otimes_k \overline{k})$$ for every $l$-adic sheaf $\mathcal{F}$ and its pullback $\mathcal{F} \otimes_k \overline{k}$ to $X \otimes_k \overline{k}$.

Think topologically and remember how people thought about sheaves in the beginning days. Well, sheaves are actually sheaves of sections of etale spaces (by this, I really mean we have some equivalence of categories), the same thing happens here: for every $l$-adic sheaf $\mathcal{F}$ on $X$, there exists an algebraic space (which plays the role of an etale space in the topological world) $[\mathcal{F}]$ together with an etale morphism $f: [\mathcal{F}] \longrightarrow X$ such that $\mathcal{F}$ becomes the sheaf of sections of this morphism. As a consequence, we may identify $\mathcal{F}$ with $[\mathcal{F}]$. By base change, we obtain an etale morphism $f \otimes_k \overline{k}: [\mathcal{F}] \otimes_k \overline{k} \longrightarrow X \otimes_k \overline{k}$ and in a similar to the theorem above, the diagram

$$\require{AMScd}
\begin{CD}
\overline{\mathcal{F}}= [\mathcal{F}] \otimes_k \overline{k} @>{\mathrm{Frob}}>>   [\mathcal{F}] \otimes_k \overline{k} \\
@V f VV @V f VV \\
 X  \otimes_k \overline{k}  @>{\mathrm{Frob}}>> X \otimes_k \overline{k}.
\end{CD}$$

is cartesian. That being said, $\overline{\mathcal{F}} \simeq  \mathrm{Frob}^*\overline{\mathcal{F}}$ where by $\mathrm{Frob}^*$ I really mean pullback of a sheaf. This isomorphism yields two important facts:

• The composition $$\mathrm{Frob}^*: H_c^i(X \otimes_k \overline{k}, \overline{\mathcal{F}}) \longrightarrow H_c^i(X \otimes_k \overline{k},\mathrm{Frob}^*\overline{\mathcal{F}}) \simeq H_c^i(X \otimes_k \overline{k}, \overline{\mathcal{F}})$$ is the one that we are seeking, where the first morphism is the natural morphism. 
• If $x \in X \otimes_k \overline{k}$ is fixed by the $n$-iteration of the absolute Frobenius, then taking stalks induces an isomorphism $\mathrm{Frob}_x^{*n}: \mathcal{F}_x \overset{\sim}{\longrightarrow} \mathcal{F}_x$.
  

Theorem 7 (Grothendieck-Lefschetz trace formula). With these data, we have

$$\sum_{x \in X(k_n)}\mathrm{Trace}(\mathrm{Frob}_x^{*n},\mathcal{F}_x) = \sum_i (-1)^i\mathrm{Trace}(\mathrm{Frob}^{*n},H^i_c(X \otimes_k \overline{k},\overline{\mathcal{F}})).$$ In particular, 

$$\sum_{x \in X(k)}\mathrm{Trace}(\mathrm{Frob}_x^{*},\mathcal{F}_x) = \sum_i (-1)^i\mathrm{Trace}(\mathrm{Frob}^{*},H^i_c(X \otimes_k \overline{k},\overline{\mathcal{F}})).$$

If we set

$$\mathrm{Trace}_{\mathcal{F}}(x) =  \mathrm{Trace}(\mathrm{Frob}_x^{*},\mathcal{F}_x)$$ for each $x \in X(k)$, then this constitues a function

$$\mathrm{Trace}: X(k) \longrightarrow \overline{\mathbb{Q}}_l = \mathbb{C}$$ with the following properties

• For any $x \in X_0(k)$ and $\mathcal{F},\mathcal{G} \in D^b_c(X)$ $$\mathrm{Trace}_{\mathcal{F}}(x)\mathrm{Trace}_{\mathcal{G}}(x) = \mathrm{Trace}_{\mathcal{F} \otimes \mathcal{G}}(x).$$
• For any morphism of $k$-varieties $f: X \longrightarrow Y$ $$\mathrm{Trace}_{f^*\mathcal{F}}(x)  = \mathrm{Trace}_{\mathcal{F}}(f(x)).$$
• For any $y \in Y(k)$ then $$\sum_{x \in X_y(k)} \mathrm{Trace}_{\mathcal{F}}(x) = \mathrm{Trace}_{f_!\mathcal{F}}(y).$$

Katz's point of view

Given a connected variety $X/k$ and for any point $x: k_r \longrightarrow X$, we get an induced group homomorphism

$$x_*: \pi_1(k_r,\overline{k}) \longrightarrow \pi_1(X,\overline{k})$$ by the functoriality of the etale fundamental group functor. Since $\pi_1(k_r)$ contains the Frobenius automorphism $\mathrm{Frob}_{k_r}: \overline{k} \longrightarrow \overline{k}, a \mapsto a^{\left| k_r \right|}$, we can consider its image via $x_*$ and set

$$x_*(\mathrm{Frob}_{k_r}) = \mathrm{Frob}_{k_r,x}.$$ Now given a smooth $l$-adic sheaf, i.e. a finitely dimensional representation 

$$\mathcal{F}: \pi_1(X) \longrightarrow \mathrm{GL}(r,\overline{\mathbb{Q}}_l),$$ and a $k$-point $x: k \longrightarrow X$ then it makes sense to consider the trace of the automorphism $\mathrm{Trace}(\mathcal{F}(\mathrm{Frob}_{k,x}))$ which is nothing but $\mathrm{Trace}_{\mathcal{F}}(x)$ considered before. However, I do not have any reference for this.

Artin-Schreier theory

Now with the formalism of $l$-adic cohomology in hands, we are ready to translate functions to cohomology. We introduce things called Artin-Schreier sheaf on $\mathbb{A}^1$ and Kummer sheaf on $\mathbb{G}_m$. Here again, $k = \mathbb{F}_q, q = p^m$.

The Artin-Schreier sheaf is the morphism 

$$\begin{align*} L: \mathbb{A}^1_k &  \longrightarrow \mathbb{A}^1_k \\ t & \longmapsto t - t^q  \end{align*}$$ (here $t$ denotes the canonical coordinate on $\mathbb{A}^1$) is an etale covering whose whose Galois group is $\mathbb{F}_q$, i.e. $\mathrm{Aut}_{\mathbb{A}^1}(\mathbb{A}^1) = k$ and generated by $x \longmapsto x+1$.  Note that the fundamental group $\pi_1(\mathbb{A}^1)$ and contains $\mathrm{Aut}_{\mathbb{A}^1}(\mathbb{A}^1)$ as an element of the projective system, so there is a canonical projection 

$$\pi_1(\mathbb{A}_k^1) \longrightarrow k.$$ Given any additive character $\psi: k  \longrightarrow \overline{\mathbb{Q}}_l^{\times}$, one then has a local system of rank $1$ from the composition 

$$\mathcal{L}_{\psi}: \pi_1(\mathbb{A}_k^1) \longrightarrow k \overset{\psi}{\longrightarrow} \overline{\mathbb{Q}}_l^{\times}$$ denoted $\mathcal{L}_{\psi}$, called the Artin-Schreier sheaf of $\psi$. The important fact is that

Lemma 8. $\mathrm{Trace}_{\mathcal{L}_{\psi}}(x)  = \psi(x)$ for any $x \in k = \mathbb{A}^1_k(k)$.

Proof. Since $\mathrm{Trace}_{\mathcal{L}_{\psi}}(x) = \mathrm{Trace}(\psi(\mathcal{L}_{\psi}(\mathrm{Frob}_{k,x})))$, we need to know what is $\mathcal{L}_{\psi}(\mathrm{Frob}_{k,x})$; in other words, where the Frobenius goes. We are done if we can prove that $ \mathcal{L}_{\psi}(\mathrm{Frob}_{k,x})=x$. To be continued.

Now we come to the main point, namely, the cohomological expression of Kloosterman sums. For any value $a$, we consider the hyperbol

$$X_a = \left \{(x,y) \in \mathbb{A}^2_k \mid xy = a \right \}$$

and consider the morphism $h_a: X_a \longrightarrow \mathbb{A}^1, (x,y) \mapsto x+y$. By theorem 7 and lemma 8, we have

$$\mathrm{Kl}(a,\psi) = \sum_{i=0}^2 (-1)^i \mathrm{Trace}(\mathrm{Frob}^*, H^i_c(X_a \otimes_k \overline{k}, h_a^*\mathcal{L}_{\psi} \otimes_k \overline{k})).$$ Note that $X_a$ is non-compact curve, so $H^0(X_a) = 0$ and by Poincare duality $H^2(X_a)=0$, therefore 

$$\mathrm{Kl}(a,\psi) = - \mathrm{Trace}(\mathrm{Frob}^*, H^1_c(X_a \otimes_k \overline{k}, h_a^*\mathcal{L}_{\psi} \otimes_k \overline{k})).$$ Note that, $$\dim \ H^1_c(X_a \otimes_k \overline{k}, h_a^*\mathcal{L}_{\psi} \otimes_k \overline{k}) = -\chi_c(X_a, h^*\mathcal{L}_{\psi})$$ the Euler characteristic with compact support. We'd like to compute this dimension first. Thanks to the Grothendieck-Ogg-Shafarevich theorem, we can compute this characteristic as follows.

Theorem 9 (Grothendieck-Ogg-Shafarevich). Let $\overline{X}$ be a proper smooth curve over $k$ and $X$ an open subset of $\overline{X}$ and $\mathcal{F}$ a local system on $X$. Then

$$\chi_c(X \otimes_k \overline{k},\mathcal{F}) = \chi_c(X \otimes_k \overline{k})\mathrm{rank}(\mathcal{F})  - \sum_{x \in \overline{X}\setminus X} \mathrm{Sw}_x(\mathcal{F})$$ where $\mathrm{Sw}$ are Swan conductors.

The Swan conductors are hard to be defined but in practice, one just needs to know its formal properties:

• $\mathrm{Sw}_x(\mathcal{F})$ depends only on its restriction to the punctured formal disc $\hat{X}_x^{\bullet}$. 
• $\mathrm{Sw}_x(\mathcal{F})=0$ when the restriction of $\mathcal{F}$ to $\hat{X}_x^{\bullet}$ is tame.
• If $\mathcal{G}$ is a tame local system at $\hat{X}_x^{\times}$, then $\mathrm{Sw}_x(\mathcal{F} \otimes \mathcal{G}) = \mathrm{Sw}_x(\mathcal{F})\mathrm{rank}(\mathcal{G}).$

Here are some computations. 

Example 10. If $X = \mathbb{A}^1$ and $\mathcal{F} = \mathcal{L}_{\psi}$, then by an elementary argument, we see that

$$\mathrm{Trace}(\mathrm{Frob}^*,H_c^1(\mathbb{A}^1,\mathcal{L}_{\psi})) = \sum_{x \in k} \psi(x) = 0$$ and hence $\chi_c(\mathbb{A}^1,\mathcal{L}_{\psi})=0$. By Grothendieck-Ogg-Shafarevich formula, we see that

$$0 = \chi_c(\mathbb{A}^1)\mathrm{rank}(\mathcal{L}_{\psi}) - \mathrm{Sw}_{\infty}(\mathcal{L}_{\psi})$$ and from this we deduce that $\mathrm{Sw}_{\infty}(\mathcal{L}_{\psi}) = 1$.

Example 11. For each $a \neq 0$, we see that $X_a \simeq \mathbb{G}_m$ so by Grothendieck-Ogg-Shafarevich formula,

$$\chi_c(X_a,h_a^*\mathcal{L}_{\psi}) = \chi_c(X_a) - \mathrm{Sw}_0(h_a^*\mathcal{L}_{\psi}) - \mathrm{Sw}_{\infty}(h_a^*\mathcal{L}_{\psi}) = - \mathrm{Sw}_0(h_a^*\mathcal{L}_{\psi}) - \mathrm{Sw}_{\infty}(h_a^*\mathcal{L}_{\psi}).$$ By the properties of Swan conductors

$$\begin{align*} \mathrm{Sw}_0(h_a^*\mathcal{L}_{\psi}) & = \mathrm{Sw}_0(x^*\mathcal{L}_{\psi} \otimes y^*\mathcal{L}_{\psi}) \\ & = \mathrm{Sw}_0(x^*\mathcal{L}_{\psi})\mathrm{rank}(y^*\mathcal{L}_{\psi})  = 1 \end{align*}$$ since $y^*\mathcal{L}_{\psi}$ is even unramified (not just tame) and by the previous example. By symmetry, $\mathrm{Sw}_{\infty}(h_a^*\mathcal{L}_{\psi}) = 1$ and finally this all implies that $\chi_c(X_a,h_a^*\mathcal{L}_{\psi})=2$.

Weight theory of Deligne

We fix once for all an identification $\iota: \overline{\mathbb{Q}}_l  \overset{\sim}{\longrightarrow} \mathbb{C}$ so that we can speak of an absolute on $\overline{\mathbb{Q}}_l$. Given a smooth $\mathbb{Q}_l$-sheaf $\mathcal{F}$ on an algebraic variety $X/k$, $k_n/k$ a finite extension of $k$.

$$\mathcal{F}: \pi_1(X) \longrightarrow \mathrm{GL}(r,\mathbb{C})$$ and a point $x \in X(k_n)$, then we say that

• $\mathcal{F}$ is pure of weight $w$ if for each $n$, every eigenvalue of $\mathrm{Frob}^{*n}_x$ has eigenvalues with absolute values $\left|k \right|^{w/2}$.
• $\mathcal{F}$ is mixed of weight $\geq w$ if if for each $n$, every eigenvalue of $\mathrm{Frob}^{*n}_x$ has eigenvalues with absolute values $\geq \left|k \right|^{w/2}$.
• $\mathcal{F}$ is mixed of weight $\leq w$ if if for each $n$, every eigenvalue of $\mathrm{Frob}^{*n}_x$ has eigenvalues with absolute values $\leq \left|k \right|^{w/2}$.

We call the celebrated theorem due to Deligne, originally known as Weil conjectures.

Target theorem 11 (Deligne). Let $X/k$ be a variety and $\mathcal{F}$ is a $l$-adic sheaf mixed of weight $\leq 0$, then every eigenvalue of 

$$\mathrm{Frob}^*:H_c^i(X \otimes_k \overline{k},  \mathcal{F} \otimes_k \overline{k}) \longrightarrow H_c^i(X \otimes_k \overline{k},  \mathcal{F} \otimes_k \overline{k})$$ has absolute values $\leq \left |k \right|^{i/2}$. 

In Weil II, Deligne proved something much stronger where one replaces $U \longrightarrow \mathrm{Spec}(k)$ by a morphism $f: X \longrightarrow Y$, then $R^if_!\mathcal{F}$ is mixed of weight $\leq w +i$  whenever $\mathcal{F}$ is mixed of weight $\leq w$. However, the Target theorem is enough to deduce the last part of the Weil conjectures and estimates of Kloosterman sums. 

From Deligne's weight theorems, the computation $\dim \ H^1_c(X_a \otimes_k \overline{k} ,h_a^*\mathcal{L}_{\psi} \otimes_k \overline{k}) = 2$, and 

$$\mathrm{Kl}(a,\psi) = - \mathrm{Trace}(\mathrm{Frob}^*, H^1_c(X_a \otimes_k \overline{k}, h_a^*\mathcal{L}_{\psi} \otimes_k \overline{k})).$$ we see that

$$\left | \mathrm{Kl}(a,\psi) \right| \leq 2p^{1/2}.$$

Moreover, if we define a generalized Kloosterman sum as 

$$\mathrm{Kl}_m(a,\psi) = \sum_{x_1\cdots x_m = a, x_i \in k}\psi(x_1 + \cdots + x_m)$$ then we have an estimate $\left |\mathrm{Kl}_m(a,\psi) \right | \leq mp^{(m-1)/2}$.