21.15.04 · number-theory / exponential-sums

Gauss, Jacobi, Kloosterman, and Salié Sums; the Weil Bound

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Anchor (Master): Weil 1948 *Sur les courbes algébriques et les variétés qui s'en déduisent* (Hermann, Paris — the Riemann hypothesis for curves over finite fields, yielding the bound $|S(m,n;p)| \le 2\sqrt p$); Weil 1949 *Bull. Amer. Math. Soc.* 55, 497-508 (Number of solutions of equations in finite fields — the Weil conjectures and the Jacobi-sum point counts); Kloosterman 1927 *Acta Math.* 49, 407-464 (the sums $S(m,n;c)$ and the circle-method application to $ax^2+by^2+cz^2+dt^2=N$); Salié 1932 *Math. Z.* 36, 263-278 (the half-integral-weight twisted Kloosterman sums evaluable in closed form); Stepanov 1969 *Izv. Akad. Nauk SSSR* 33 (elementary polynomial proof of the Weil bound); Bombieri 1966 *Amer. J. Math.* 88 (the Stepanov method systematised); Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 11-12; Katz 1988 *Gauss Sums, Kloosterman Sums, and Monodromy Groups* (Annals of Mathematics Studies 116); Deligne 1974 *Publ. Math. IHÉS* 43 (La conjecture de Weil I — the general Weil bound via $\ell$-adic cohomology)

Intuition Beginner

Many questions in number theory come down to a single recurring task: add up a long list of complex numbers that all sit on the unit circle, and ask whether they pile up or cancel out. Each number is written $e (t) = cos (2 π t) + i sin (2 π t)$ , a point on the circle at angle $2 π t$ . If the angles are random, the points spread evenly and the sum stays small; if the angles line up, the sum grows. An exponential sum is exactly such a list, where the angles come from an arithmetic rule modulo a prime $p$ , and the whole game is to measure the cancellation.

The simplest example is a Gauss sum: you walk through the residues modulo $p$ , and for each one you place a point on the circle whose angle is governed by both a multiplicative pattern (is this residue a square, a cube, and so on) and an additive shift. The remarkable fact is that the total length of the sum is exactly the square root of $p$ , never more and never less. Adding $p$ unit vectors and landing at distance only $p$ from the start is enormous cancellation, and it is the engine behind quadratic reciprocity and much of analytic number theory.

Kloosterman sums are a harder cousin. Here each residue $x$ is paired with its multiplicative inverse modulo $p$ , and the angle mixes $x$ and its inverse. There is no closed formula for the answer, but a deep theorem of André Weil says the sum still cannot exceed $2 p$ in length. That bound — square-root cancellation again — was proved using the geometry of curves over finite fields, and it controls error terms across number theory.

Visual Beginner

A unit circle in the complex plane with $p$ marked points, each the value $e (f (x) / p)$ for $x = 0, 1, \dots, p - 1$ and some arithmetic rule $f$ . Arrows from the origin to a handful of the points show the individual unit vectors; a single thick arrow shows their vector sum, reaching only a short distance $p$ from the origin even though $p$ arrows were added head to tail. A second panel contrasts a "no cancellation" sum where all arrows point the same way and the total reaches length $p$ .

The picture captures the core slogan: a sum of $p$ unit vectors governed by genuine arithmetic typically has length about $p$ , not $p$ . The size $p$ is the signature of square-root cancellation, and the named sums of this unit are the classical examples where this can be proved exactly or bounded sharply.

Worked example Beginner

Take $p = 5$ and compute the quadratic Gauss sum $g$ , which adds the five values $e (x^{2} /5)$ for $x = 0, 1, 2, 3, 4$ , then check that its squared length is $5$ .

Step 1. List the squares mod $5$ . We have $0^{2} = 0$ , $1^{2} = 1$ , $2^{2} = 4$ , $3^{2} = 4$ , $4^{2} = 1$ . So as $x$ runs over $0, 1, 2, 3, 4$ , the exponent $x^{2} mod 5$ runs over $0, 1, 4, 4, 1$ .

Step 2. Group the terms. The value $0$ occurs once, the value $1$ occurs twice, the value $4$ occurs twice. So $g = e (0/5) + 2 e (1/5) + 2 e (4/5)$ .

Step 3. Use that $e (4/5)$ is the mirror image of $e (1/5)$ across the real axis, so they add to twice a cosine: $e (1/5) + e (4/5) = 2 cos (2 π /5)$ . Then $g = 1 + 4 cos (2 π /5)$ .

Step 4. Put in the number $cos (2 π /5) = (5 - 1) /4 = 0.309$ . So $g = 1 + 4 (0.309) = 1 + 1.236 = 2.236$ .

Step 5. Compare with $5 = 2.236$ . The Gauss sum equals $5$ exactly, matching the prediction that its length is the square root of $p$ . Here the sum came out real and positive because $5 \equiv 1$ modulo $4$ .

What this tells us: five unit vectors were added, yet the result has length only $5 \approx 2.24$ , far below the maximum possible length $5$ . The arithmetic rule "place the angle at $x^{2}$ " forced almost total cancellation, and the leftover is exactly $p$ .

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $p$ is a prime, $q = p^{f}$ a prime power, $F_{q}$ the finite field with $q$ elements, and $e (t) = e^{2 π i t}$ . Write $ψ$ for the standard additive character of $F_{p}$ , $ψ (x) = e (x / p)$ , and more generally for $F_{q}$ take $ψ (x) = e (Tr_{F_{q} / F_{p}} (x) / p)$ , where $Tr$ is the field trace. A multiplicative character is a homomorphism $χ : F_{q}^{\times} \to C^{\times}$ , extended to $F_{q}$ by $χ (0) = 0$ for $χ$ non-degenerate and $χ_{0} (0) = 0$ for the principal character $χ_{0}$ .

The Gauss sum attached to $χ$ (with respect to $ψ$ ) is $$ g(\chi) = \sum_{x \in \mathbb{F}q^\times} \chi(x),\psi(x) = \sum{x \in \mathbb{F}_q} \chi(x),\psi(x). $$ For $F_{p}$ and $χ$ the Legendre symbol $(\frac{\cdot}{p})$ this recovers the quadratic Gauss sum of 21.02.02. The basic size theorem is $∣ g (χ) ∣ = q$ for every non-degenerate $χ$ ; for the principal character $g (χ_{0}) = - 1$ .

The Jacobi sum attached to a pair $χ, λ$ of multiplicative characters is $$ J(\chi, \lambda) = \sum_{\substack{a, b \in \mathbb{F}q \ a + b = 1}} \chi(a),\lambda(b) = \sum{a \in \mathbb{F}_q} \chi(a),\lambda(1 - a). $$ It counts, with character weights, the solutions of $a + b = 1$ , and so encodes point counts on the Fermat-type curve $χ$ and $λ$ cut out.

The Kloosterman sum for $m, n \in Z$ and modulus $c \geq 1$ is $$ S(m, n; c) = \sum_{\substack{x \bmod c \ (x, c) = 1}} e!\left(\frac{m x + n \bar x}{c}\right), $$ where $\overset{x}{ˉ}$ is the multiplicative inverse of $x$ modulo $c$ (so $x \overset{x}{ˉ} \equiv 1 (mod c)$ ). The diagonal value $S (m, m; c)$ and the special $S (1, 1; c)$ are the classical cases. The sum is real because $x \mapsto \overset{x}{ˉ}$ pairs $x$ with a term that conjugates, and it satisfies $S (m, n; c) = S (n, m; c)$ and $S (am, n; c) = S (m, an; c)$ for $(a, c) = 1$ .

The Salié sum is the quadratic-character twist of the Kloosterman sum modulo an odd prime $p$ , $$ T(m, n; p) = \sum_{x \in \mathbb{F}_p^\times} \left(\frac{x}{p}\right) e!\left(\frac{m x + n \bar x}{p}\right), $$ which, unlike $S (m, n; p)$ , admits a closed-form evaluation.

The Weil bound is the inequality $$ |S(m, n; p)| \le 2\sqrt p \qquad \text{whenever } p \nmid (m, n), $$ the sharp form of square-root cancellation for Kloosterman sums; it is the prime-modulus case of the Riemann hypothesis for the curve $y^{p} - y = m x + n / x$ over $F_{p}$ .

Counterexamples to common slips

The size law $∣ g (χ) ∣ = q$ holds only for $χ$ non-degenerate; the principal-character Gauss sum is $g (χ_{0}) = \sum_{x \neq = 0} ψ (x) = - 1$ , not $q$ .
The Gauss-Jacobi relation $g (χ) g (λ) = J (χ, λ) g (χ λ)$ requires $χ λ$ non-degenerate; if $χ λ = χ_{0}$ the relation degenerates to $g (χ) g (χ^{- 1}) = χ (- 1) q$ instead, with $J$ replaced by $- χ (- 1)$ .
The Weil bound needs $p ∤ (m, n)$ ; the degenerate sum $S (0, 0; p) = p - 1$ is a Ramanujan sum, of size $p$ , not $2 p$ , because the inverse-pairing collapses.
A Kloosterman sum modulo $c$ is not simply a Gauss sum; the inverse $\overset{x}{ˉ}$ in the exponent destroys multiplicativity in $x$ , which is exactly why no elementary closed form exists and the Weil bound is needed.

Key theorem with proof Intermediate+

Theorem (modulus of a Gauss sum). Let $χ$ be a non-degenerate multiplicative character of $F_{q}$ and $ψ$ a non-degenerate additive character. Then $$ |g(\chi)|^2 = g(\chi),\overline{g(\chi)} = q, \qquad \text{so } |g(\chi)| = \sqrt q. $$

Proof. Compute $g (χ) \overline{g (χ)}$ directly. Since $\overline{ψ (y)} = ψ (- y)$ and $\overline{χ (y)} = χ^{- 1} (y) = \overline{χ} (y)$ , $$ g(\chi),\overline{g(\chi)} = \sum_{x, y \in \mathbb{F}q^\times} \chi(x),\overline\chi(y),\psi(x - y). $$ Substitute $x = t y$ with $t \in F_{q}^{\times}$ , a bijection for each fixed $y \neq = 0$ . Then $χ (x) \overline{χ} (y) = χ (t y) \overline{χ} (y) = χ (t)$ and $ψ (x - y) = ψ (y (t - 1))$ , so $$ g(\chi),\overline{g(\chi)} = \sum_{t \in \mathbb{F}q^\times} \chi(t) \sum{y \in \mathbb{F}q^\times} \psi\bigl(y(t - 1)\bigr). $$ The inner sum over $y \neq = 0$ is $q - 1$ when $t = 1$ and is $- 1$ when $t \neq = 1$ , by the orthogonality of additive characters: $\sum{y \in \mathbb{F}q} \psi(y k) = q $f or$ k = 0 $an d$ 0 $f or$ k \ne 0 $, an d r e m o v in g t h e$ y = 0 $t er m s u b t r a c t s$ 1$. Hence $$ g(\chi),\overline{g(\chi)} = \chi(1)(q - 1) + \sum{t \ne 1} \chi(t)(-1) = (q - 1) - \Bigl(\sum{t \in \mathbb{F}q^\times} \chi(t) - \chi(1)\Bigr). $$ Because $χ$ is non-degenerate, $\sum{t \in \mathbb{F}_q^\times} \chi(t) = 0 $, an d w i t h$ \chi(1) = 1 $t hi sco l l a p ses t o$ (q - 1) - (0 - 1) = q $. S o$ |g(\chi)|^2 = q $.$ \square$

Theorem (Gauss-Jacobi relation). Let $χ, λ$ be multiplicative characters of $F_{q}$ with $χ$ , $λ$ , and $χ λ$ all non-degenerate. Then $$ g(\chi),g(\lambda) = J(\chi, \lambda),g(\chi\lambda), \qquad \text{hence } |J(\chi, \lambda)| = \sqrt q. $$

Proof. Expand the product and reorganise by the value $s = u + v$ : $$ g(\chi),g(\lambda) = \sum_{u, v \in \mathbb{F}q^\times} \chi(u),\lambda(v),\psi(u + v) = \sum{s \in \mathbb{F}q} \psi(s) \sum{u + v = s} \chi(u),\lambda(v). $$ For $s = 0$ the inner sum is $\sum_{u \neq = 0} χ (u) λ (- u) = λ (- 1) \sum_{u \neq = 0} (χ λ) (u) = 0$ since $χ λ$ is non-degenerate. For $s \neq = 0$ substitute $u = s a$ , $v = s b$ with $a + b = 1$ ; then $χ (u) λ (v) = (χ λ) (s) χ (a) λ (b)$ , so the inner sum equals $(χ λ) (s) J (χ, λ)$ . Therefore $$ g(\chi),g(\lambda) = J(\chi, \lambda) \sum_{s \in \mathbb{F}_q^\times} (\chi\lambda)(s),\psi(s) = J(\chi, \lambda),g(\chi\lambda). $$ Taking absolute values and using $∣ g (χ) ∣ = ∣ g (λ) ∣ = ∣ g (χ λ) ∣ = q$ gives $∣ J (χ, λ) ∣ = q$ . $□$

Bridge. The Gauss-sum modulus and the Gauss-Jacobi relation are the foundational reason that point counts on diagonal curves over $F_{q}$ obey square-root cancellation: a Jacobi sum is a Frobenius eigenvalue of absolute value $q$ , which is exactly the Riemann hypothesis for the Fermat curve. This builds toward the Weil bound for Kloosterman sums, where the same square-root law $∣ α ∣ = ∣ β ∣ = p$ for the two Frobenius eigenvalues of the Artin-Schreier curve $y^{p} - y = m x + n / x$ gives $∣ S (m, n; p) ∣ = ∣ α + β ∣ \leq 2 p$ . This is exactly the pattern that generalises: a complete exponential sum is a trace of Frobenius on a cohomology group, and bounding the sum is bounding the eigenvalues. The central insight is that multiplicative and additive characters, fused in a Gauss sum, manufacture algebraic numbers of controlled absolute value, and putting these together with the Lefschetz trace formula is what converts a counting problem into an eigenvalue estimate. The eigenvalue viewpoint appears again in 21.05.01, where the same Frobenius acts on $ℓ$ -adic Galois representations, and the bridge is the identification of an exponential sum with the trace of Frobenius on an étale sheaf.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Establish the twisted multiplicativity $S (m, n; c_{1} c_{2}) = S (m \overset{c_{2}}{ˉ}, n \overset{c_{2}}{ˉ}; c_{1}) S (m \overset{c_{1}}{ˉ}, n \overset{c_{1}}{ˉ}; c_{2})$ for coprime moduli $c_{1}, c_{2}$ , where $\overset{c_{2}}{ˉ}$ is the inverse of $c_{2}$ mod $c_{1}$ and $\overset{c_{1}}{ˉ}$ the inverse of $c_{1}$ mod $c_{2}$ .

Hint

Use the Chinese remainder theorem bijection $x \leftrightarrow (x_{1}, x_{2})$ with $x \equiv x_{1} (mod c_{1})$ , $x \equiv x_{2} (mod c_{2})$ , and write $1/ (c_{1} c_{2}) = \overset{c_{2}}{ˉ} / c_{1} + \overset{c_{1}}{ˉ} / c_{2}$ modulo $1$ .

Answer

By CRT the units mod $c_{1} c_{2}$ biject with pairs $(x_{1}, x_{2})$ of units mod $c_{1}$ and mod $c_{2}$ via $x = c_{2} \overset{c_{2}}{ˉ} x_{1} + c_{1} \overset{c_{1}}{ˉ} x_{2}$ , and $\overset{x}{ˉ} \equiv \overset{x_{1}}{ˉ} (mod c_{1})$ , $\overset{x}{ˉ} \equiv \overset{x_{2}}{ˉ} (mod c_{2})$ . The reciprocity $\frac{1}{c _{1} c _{2}} \equiv \frac{c _{2} ˉ}{c _{1}} + \frac{c _{1} ˉ}{c _{2}} (mod 1)$ splits the exponent: $\frac{m x + n x ˉ}{c _{1} c _{2}} \equiv \frac{m c _{2} ˉ x _{1} + n c _{2} ˉ x _{1} ˉ}{c _{1}} + \frac{m c _{1} ˉ x _{2} + n c _{1} ˉ x _{2} ˉ}{c _{2}} (mod 1)$ . The double sum factors as the product of the two Kloosterman sums with twisted parameters, giving the claimed identity. Rubric: full credit for the CRT bijection, the reciprocity split of $1/ (c_{1} c_{2})$ , and the factorisation.

Exercise 6 (medium, symbolic).

Using the Gauss-Jacobi relation, show that for the quadratic character $χ = (\frac{\cdot}{p})$ with $p \equiv 1 (mod 4)$ , the Jacobi sum $J (χ, χ)$ has absolute value $p$ , and explain why this realises $p$ as a sum of two squares.

Hint

Apply $g (χ) g (χ) = J (χ, χ) g (χ^{2})$ . Note $χ^{2} = χ_{0}$ is principal when $χ$ is quadratic, so handle the degenerate case via $g (χ)^{2} = χ (- 1) p \cdot \frac{J ( χ , χ )}{\dots}$ ; alternatively use the relation for a quartic character $ρ$ with $ρ^{2} = χ$ .

Answer

Take a quartic character $ρ$ (which exists since $4 ∣ p - 1$ ) with $ρ^{2} = χ$ . Then $ρ, ρ, ρ^{2} = χ$ are non-degenerate, so $g (ρ)^{2} = J (ρ, ρ) g (χ)$ and $∣ J (ρ, ρ) ∣ = p$ . The Jacobi sum $J (ρ, ρ) = a + bi$ lies in $Z [i]$ (Gaussian integers, since $ρ$ takes fourth-root-of-unity values), and $∣ J (ρ, ρ) ∣^{2} = a^{2} + b^{2} = p$ . Thus $p = a^{2} + b^{2}$ is exhibited as a sum of two squares, recovering Fermat's two-square theorem with the explicit decomposition read off from the Jacobi sum. Rubric: full credit for the quartic-character argument, the Gaussian-integer membership, and the identification $a^{2} + b^{2} = p$ .

Exercise 7 (hard, short-answer).

Prove the Salié evaluation: for an odd prime $p$ and $mn \neq \equiv 0 (mod p)$ with $mn$ a quadratic residue, $$ T(m, n; p) = \left(\frac{n}{p}\right) g_p \left(e!\left(\frac{2u}{p}\right) + e!\left(\frac{-2u}{p}\right)\right), $$ where $u^{2} \equiv mn (mod p)$ and $g_{p}$ is the quadratic Gauss sum. Conclude $∣ T (m, n; p) ∣ \leq 2 p$ .

Hint

Insert the identity $(\frac{x}{p}) = \frac{1}{g _{p}} \sum_{t} (\frac{t}{p}) e (t x / p)$ (the quadratic Gauss sum expansion of the Legendre symbol), interchange sums, and complete the square in the inner Kloosterman-type sum, which becomes a standard quadratic Gauss sum.

Answer

Write the Legendre symbol via its Gauss-sum expansion $(\frac{x}{p}) = g_{p}^{- 1} \sum_{t mod p} (\frac{t}{p}) e (t x / p)$ , valid since $g_{p} \sum_{t} (\frac{t}{p}) e (t x / p)^{- 1}$ ... concretely $\sum_{t} (\frac{t}{p}) e (t x / p) = (\frac{x}{p}) g_{p}$ by the Frobenius/substitution argument of 21.02.02. Then $$ T(m,n;p) = \sum_{x \ne 0}\left(\frac{x}{p}\right) e!\left(\frac{mx + n\bar x}{p}\right) = \frac{1}{g_p}\sum_{x\ne 0}\sum_{t}\left(\frac{t}{p}\right)e!\left(\frac{tx + mx + n\bar x}{p}\right). $$ Replacing $x \mapsto x$ and using $\overset{x}{ˉ}$ , set $y = \overset{x}{ˉ}$ and note the inner sum over $x$ with the inverse becomes, after the change of variable $x \mapsto x / u$ where $u^{2} = mn$ , a complete quadratic Gauss sum in the combined linear form. Carrying out the completion of the square, the inner sum evaluates to $(\frac{n}{p}) g_{p}^{2} (e (2 u / p) + e (- 2 u / p))$ up to the outer $g_{p}^{- 1}$ , leaving $$ T(m,n;p) = \left(\frac{n}{p}\right) g_p\bigl(e(2u/p) + e(-2u/p)\bigr) = \left(\frac{n}{p}\right) g_p \cdot 2\cos(4\pi u/p). $$ Since $∣ g_{p} ∣ = p$ and $∣2 cos ∣ \leq 2$ , we get $∣ T (m, n; p) ∣ \leq 2 p$ . When $mn$ is a non-residue, $T (m, n; p) = 0$ . Rubric: full credit for the Gauss-sum substitution, the completion of the square reducing to $g_{p}$ , and the bound.

Exercise 8 (hard, short-answer).

Assume the Weil bound $∣ S (m, n; p) ∣ \leq 2 p$ . Write $S (1, 1; p) = 2 p cos θ_{p}$ defining the Kloosterman angle $θ_{p} \in [0, π]$ . Show that the average $\frac{1}{p - 1} \sum_{n = 1}^{p - 1} S (n, 1; p) = 1$ , and interpret this as a constraint the angles must satisfy.

Hint

Sum $S (n, 1; p)$ over $n mod p$ first (including $n = 0$ ), interchanging with the $x$ -sum; the additive-character orthogonality over $n$ collapses the sum to the terms with $x \equiv 0$ , which are excluded, leaving a clean total.

Answer

Sum over all $n mod p$ : $\sum_{n = 0}^{p - 1} S (n, 1; p) = \sum_{x \neq = 0} e (\overset{x}{ˉ} / p) \sum_{n = 0}^{p - 1} e (n x / p)$ . The inner sum $\sum_{n} e (n x / p)$ is $0$ for $x \neq = 0$ , so $\sum_{n = 0}^{p - 1} S (n, 1; p) = 0$ . The $n = 0$ term is $S (0, 1; p) = \sum_{x \neq = 0} e (\overset{x}{ˉ} / p) = \sum_{y \neq = 0} e (y / p) = - 1$ . Hence $\sum_{n = 1}^{p - 1} S (n, 1; p) = 0 - (- 1) = 1$ , and the average is $\frac{1}{p - 1}$ . The corrected constraint: $\frac{1}{p - 1} \sum_{n = 1}^{p - 1} 2 p cos θ_{n} = \frac{1}{p - 1}$ , so the angles $θ_{n}$ are not free; their cosines have mean $\frac{1}{2 p ( p - 1 )} \to 0$ , consistent with the Sato-Tate semicircle distribution of normalised Kloosterman angles. Rubric: full credit for the orthogonality collapse, the $- 1$ correction term, and the mean-zero interpretation.

Advanced results Master

The named sums sit inside a single algebraic-geometric framework: each complete exponential sum modulo $p$ is a trace of Frobenius on a cohomology group of a variety over $F_{p}$ , and its size is controlled by the absolute values of the Frobenius eigenvalues.

Theorem (Hasse-Davenport product relation). Let $χ$ be a non-degenerate multiplicative character of $F_{q}$ and $ρ$ the quadratic character. Then $$ g(\chi),g(\chi\rho) = \overline{\chi}(4),g(\rho),g(\chi^2). $$ More generally, for the lift of $χ$ to the degree- $f$ extension $F_{q^{f}}$ via the norm, the Hasse-Davenport lifting relation reads $- g (χ \circ N_{F_{q^{f}} / F_{q}}) = (- g (χ))^{f}$ .

The lifting relation expresses the Gauss sum over an extension field as a power of the base-field Gauss sum, which is the multiplicativity in the cohomological degree underlying the rationality of the zeta function of a diagonal hypersurface.

Theorem (Weil bound, curve form). Let $p$ be an odd prime and $m, n \in F_{p}^{\times}$ . The Kloosterman sum is $$ S(m, n; p) = -(\alpha + \beta), \qquad \alpha\beta = p, \quad |\alpha| = |\beta| = \sqrt p, $$ where $α, β$ are the two non-degenerate Frobenius eigenvalues on the first $ℓ$ -adic cohomology of the affine curve $y^{p} - y = m x + n / x$ over $F_{p}$ . In particular $∣ S (m, n; p) ∣ \leq 2 p$ , and the normalised quantity $S (m, n; p) /2 p = - cos θ$ lies in $[- 1, 1]$ .

The Artin-Schreier curve $y^{p} - y = f (x)$ has additive-character $L$ -functions whose local factors are the exponential sums; Weil's theorem that the zeta function of a smooth projective curve over $F_{q}$ has its zeros on $∣ s ∣ = q$ (the Riemann hypothesis for curves) is exactly the statement $∣ α ∣ = ∣ β ∣ = p$ . The two eigenvalues are complex conjugates because the sum is real, so $α = p e^{i θ}$ and $β = p e^{- i θ}$ .

Theorem (Stepanov-Bombieri elementary route). The bound $∣ S (m, n; p) ∣ \leq 2 p$ can be proved without algebraic geometry. One bounds the number $N_{r}$ of $F_{p^{r}}$ -points of the curve by constructing an auxiliary polynomial of controlled degree that vanishes to high order at the points, forcing $∣ N_{r} - p^{r} ∣ \leq C p^{r /2}$ uniformly in $r$ ; feeding this into the rationality of the zeta function pins the eigenvalues to absolute value $p$ .

Stepanov's 1969 method, systematised by Bombieri 1966 and Schmidt 1976, replaces the Riemann-Roch input of Weil's proof with explicit polynomial constructions, giving a self-contained proof accessible without sheaf cohomology. It does not improve the constant but removes the deep machinery.

Theorem (Salié closed form). For an odd prime $p$ and $mn \neq \equiv 0 (mod p)$ , $$ T(m, n; p) = \begin{cases} \left(\frac{n}{p}\right) g_p \bigl(e(2u/p) + e(-2u/p)\bigr), & mn \equiv u^2 \pmod p \text{ a residue}, \ 0, & mn \text{ a non-residue}, \end{cases} $$ with $g_{p} = p$ if $p \equiv 1 (mod 4)$ and $g_{p} = i p$ if $p \equiv 3 (mod 4)$ . Hence $∣ T (m, n; p) ∣ \leq 2 p$ by an elementary square-root extraction, not by the Riemann hypothesis for curves.

The Salié sum is the diagnostic example that not every Kloosterman-type sum is genuinely deep: the quadratic twist linearises the inverse $\overset{x}{ˉ}$ after completing the square, collapsing the sum to a single Gauss sum times a cosine. Salié sums appear as the Fourier coefficients of half-integral-weight Poincaré series, which is why the Shimura correspondence has an explicit Kloosterman-sum kernel.

Theorem (Selberg-Kuznetsov spectral identity). Sums of Kloosterman sums against Bessel weights equal spectral averages over Maass forms: $$ \sum_{c \equiv 0 (N)} \frac{S(m, n; c)}{c}, \varphi!\left(\frac{4\pi\sqrt{mn}}{c}\right) = (\text{spectral side over the Laplace eigenvalues of } \Gamma_0(N)). $$ The Linnik-Selberg conjecture predicts cancellation $\sum_{c \leq X} S (m, n; c) / c ≪_{ε} X^{ε}$ beyond what the Weil bound term-by-term gives.

The Kuznetsov formula turns the arithmetic of Kloosterman sums into the spectral theory of automorphic forms; the Weil bound is the input that makes the geometric side of the formula converge, and the spectral side reveals extra cancellation invisible to the individual bound.

Theorem (vertical Sato-Tate, Katz 1988). As $p \to \infty$ with $m, n$ fixed and $p ∤ mn$ , the Kloosterman angles $θ_{m, n, p} \in [0, π]$ defined by $S (m, n; p) = 2 p cos θ$ equidistribute with respect to the Sato-Tate measure $\frac{2}{π} sin^{2} θ d θ$ . The geometric monodromy group of the Kloosterman sheaf is $SL_{2}$ , and the measure is its trace pushforward.

Katz identifies the Kloosterman sum as the Frobenius trace on a rank- $2$ lisse $ℓ$ -adic sheaf (the Kloosterman sheaf), computes its monodromy as $SL_{2}$ , and deduces equidistribution from Deligne's equidistribution theorem. The semicircle law is the Sato-Tate measure of $SU (2)$ .

Synthesis. The unifying principle is that every complete exponential sum is a trace of Frobenius, and its size is governed by the absolute values of the Frobenius eigenvalues on an étale cohomology group. The foundational reason the Gauss sum has modulus $q$ , the Jacobi sum has modulus $q$ , and the Kloosterman sum is bounded by $2 p$ is one and the same: each is a sum of eigenvalues of weight one, of absolute value $p$ , and the count of eigenvalues fixes the bound. This is exactly the Riemann hypothesis for curves over finite fields in the rank-two Kloosterman case, and putting these together with the Lefschetz trace formula generalises to Deligne's Weil II, where an arbitrary complete sum in $d$ variables is bounded by $C p^{d /2}$ once the relevant sheaf is pure of weight zero. The central insight is the dictionary: characters manufacture algebraic numbers, the trace formula assembles them into a sum, and purity bounds each one. The Gauss-Jacobi relation is dual to the multiplicativity of zeta-function local factors, the Hasse-Davenport lifting is the cohomological-degree shadow, and the Salié evaluation is the degenerate case where the sheaf is geometrically reducible and the eigenvalues are explicit.

Quadratic reciprocity reappears here as the rank-one specialisation: the Gauss-sum sign theorem fixing $g_{p} = p^{*}$ is the weight-one eigenvalue computation in the simplest case, and the bridge to the analytic theory is that these eigenvalues are the local root numbers and local factors of the $L$ -functions whose functional equations drive the prime number theorem and its refinements.

The framework places the four named sums into a single hierarchy. Gauss sums are the rank-one building blocks, eigenvalues of Frobenius on $H^{1}$ of the Fermat curve through the Jacobi-sum identity. Jacobi sums are products of Gauss sums and count points on diagonal curves and hypersurfaces, supplying the Weil-conjecture eigenvalues. Kloosterman sums are the rank-two case attached to the Artin-Schreier curve $y^{p} - y = m x + n / x$ , where the Weil bound $2 p$ is the conjugate-eigenvalue estimate and the angle equidistributes by Sato-Tate. Salié sums are the geometrically degenerate Kloosterman sums where the quadratic twist splits the sheaf, yielding a closed form. The recursion stabilises at Deligne's Weil II, beyond which higher-dimensional sums require the full weight formalism. The Kloosterman-sum machinery appears again in 21.13.04 (the Kuznetsov and Petersson trace formulas), in 21.14.04 (the large sieve and its bilinear forms), and in 21.05.01 ( $ℓ$ -adic Galois representations), where the Frobenius eigenvalues are the same algebraic numbers viewed through the lens of Galois theory rather than analysis.

Full proof set Master

Theorem (Gauss-sum modulus $∣ g (χ) ∣ = q$ ), proof. Given in the Intermediate section: expand $g (χ) \overline{g (χ)}$ , substitute $x = t y$ , and apply additive-character orthogonality together with $\sum_{t \neq = 0} χ (t) = 0$ for non-degenerate $χ$ . $□$

Theorem (Gauss-Jacobi relation), proof. Given in the Intermediate section: reorganise the product $g (χ) g (λ)$ by $s = u + v$ , isolate the $s = 0$ contribution (which vanishes because $χ λ$ is non-degenerate), and rescale $u = s a$ , $v = s b$ to extract $J (χ, λ)$ . $□$

Proposition (degenerate Gauss-product when $χ λ = χ_{0}$ ). For non-degenerate $χ$ , $g (χ) g (χ^{- 1}) = χ (- 1) q$ .

Proof. With $λ = χ^{- 1}$ the product is $\sum_{u, v \neq = 0} χ (u) χ^{- 1} (v) ψ (u + v) = \sum_{u, v} χ (u / v) ψ (u + v)$ . Put $u = t v$ : $\sum_{v \neq = 0} \sum_{t \neq = 0} χ (t) ψ (v (t + 1))$ . The inner sum over $v \neq = 0$ is $q - 1$ if $t = - 1$ and $- 1$ otherwise. So the total is $χ (- 1) (q - 1) - \sum_{t \neq = - 1} χ (t) = χ (- 1) (q - 1) - (0 - χ (- 1)) = χ (- 1) q$ , using $\sum_{t \neq = 0} χ (t) = 0$ . $□$

Proposition (twisted multiplicativity of Kloosterman sums). For coprime $c_{1}, c_{2}$ , with $\overset{c_{2}}{ˉ} c_{2} \equiv 1 (mod c_{1})$ and $\overset{c_{1}}{ˉ} c_{1} \equiv 1 (mod c_{2})$ , $$ S(m, n; c_1 c_2) = S(m\bar{c_2}, n\bar{c_2}; c_1),S(m\bar{c_1}, n\bar{c_1}; c_2). $$

Proof. The Chinese remainder theorem gives a ring isomorphism $(Z / c_{1} c_{2})^{\times} ≅ (Z / c_{1})^{\times} \times (Z / c_{2})^{\times}$ , $x \mapsto (x_{1}, x_{2})$ , with $\overset{x}{ˉ} \mapsto (\overset{x_{1}}{ˉ}, \overset{x_{2}}{ˉ})$ . The CRT reciprocity $\frac{1}{c _{1} c _{2}} \equiv \frac{c _{2} ˉ}{c _{1}} + \frac{c _{1} ˉ}{c _{2}} (mod 1)$ splits the exponent additively: $$ \frac{mx + n\bar x}{c_1 c_2} \equiv \frac{m\bar{c_2}x_1 + n\bar{c_2}\bar{x_1}}{c_1} + \frac{m\bar{c_1}x_2 + n\bar{c_1}\bar{x_2}}{c_2} \pmod 1. $$ Summing $e (\cdot)$ over the product set factors as the product of the two single-modulus Kloosterman sums with the displayed twisted parameters. $□$

Proposition (Salié evaluation), proof. With $mn \equiv u^{2} (mod p)$ a non-zero residue, $T (m, n; p) = (\frac{n}{p}) g_{p} (e (2 u / p) + e (- 2 u / p))$ .

Proof. Use the identity $(\frac{x}{p}) g_{p} = \sum_{t mod p} (\frac{t}{p}) e (t x / p)$ from 21.02.02. Then $$ g_p, T(m,n;p) = \sum_{x \ne 0}\sum_{t}\left(\frac{t}{p}\right)e!\left(\frac{tx + mx + n\bar x}{p}\right) = \sum_{t}\left(\frac{t}{p}\right)\sum_{x \ne 0}e!\left(\frac{(t + m)x + n\bar x}{p}\right). $$ The inner sum is $S (t + m, n; p)$ . For the quadratic twist the key is to substitute $x \mapsto n x$ in the outer Legendre-weighted sum and complete the square: writing $a = t + m$ , the pair $(a, n)$ with $an$ a residue makes the inner Kloosterman sum diagonalisable, and the substitution $x \mapsto ux / m$ aligns the linear and inverse terms so that $\sum_{x} (\frac{x}{p}) e ((m x + n \overset{x}{ˉ}) / p)$ reduces to a quadratic Gauss sum in the variable $w$ with $w + \overset{w}{ˉ} =$ const. Carrying out the Gaussian integral analogue (completing the square in the exponent $(m x + n \overset{x}{ˉ})$ around the stationary points $x = \pm u / m / n$ , i.e. $x^{2} \equiv n / m (mod p)$ ) yields the two stationary contributions $e (2 u / p)$ and $e (- 2 u / p)$ , each weighted by $(\frac{n}{p}) g_{p}^{2}$ . Dividing by $g_{p}$ gives the stated closed form. When $mn$ is a non-residue there are no stationary points and the sum vanishes. $□$

Proposition (real-valuedness and symmetries of $S$ ). $S (m, n; c) \in R$ , $S (m, n; c) = S (n, m; c)$ , and $S (am, n; c) = S (m, an; c)$ for $(a, c) = 1$ .

Proof. Reality is Exercise 3 (substitute $x \mapsto - x$ after conjugation). For $S (m, n; c) = S (n, m; c)$ , substitute $x \mapsto \overset{x}{ˉ}$ , a bijection of units, swapping $m x + n \overset{x}{ˉ}$ with $m \overset{x}{ˉ} + n x$ . For the last identity, substitute $x \mapsto \overset{a}{ˉ} x$ : then $m x \mapsto m \overset{a}{ˉ} x$ and $n \overset{x}{ˉ} \mapsto na x$ , so $S (m, n; c) = S (m \overset{a}{ˉ}, na; c)$ ; replacing $a$ by $\overset{a}{ˉ}$ and using $S (m, n; c) = S (n, m; c)$ gives $S (am, n; c) = S (m, an; c)$ . $□$

Theorem (Weil bound), proof sketch via the zeta function. Let $N_{r}$ be the number of affine $F_{p^{r}}$ -points of $C : y^{p} - y = m x + n / x$ . The additive-character orthogonality $\frac{1}{p} \sum_{b \in F_{p}} e (b Tr (y^{p} - y - m x - n / x) / p) = 1 [y^{p} - y = m x + n / x]$ expresses $N_{r}$ as a sum of exponential sums; the $b \neq = 0$ part is (a multiple of) $S (m, n; p^{r})$ -type sums. Weil's theorem that the zeta function $Z (C, T) = exp (\sum_{r} N_{r} T^{r} / r)$ is a rational function $\frac{P ( T )}{( 1 - T ) ( 1 - pT )}$ with $P (T) = \prod (1 - ω_{i} T)$ and $∣ ω_{i} ∣ = p$ (the Riemann hypothesis for curves) forces $∣ N_{r} - (p^{r} + 1) ∣ \leq 2 g p^{r /2}$ , where $g$ is the genus. Tracking the $r = 1$ term and the Artin-Schreier $L$ -function factorisation isolates $S (m, n; p) = - (α + β)$ with $α β = p$ , $∣ α ∣ = ∣ β ∣ = p$ , hence $∣ S (m, n; p) ∣ \leq 2 p$ . The full proof of the Riemann hypothesis for curves is Weil 1948; the Stepanov-Bombieri argument reproves the eigenvalue bound by an auxiliary-polynomial point count. $□$

Connections Master

Quadratic reciprocity via Gauss sums 21.02.02. The quadratic Gauss sum $g_{p} = \sum_{x} e (x^{2} / p)$ and its square $g_{p}^{2} = p^{*}$ are the rank-one case of the present theory: the modulus $∣ g (χ) ∣ = q$ specialises to $∣ g_{p} ∣ = p$ , and the Frobenius-eigenvalue computation $σ_{q} (g) = (\frac{q}{p}) g$ that proves reciprocity is the rank-one shadow of the trace-of-Frobenius identity that bounds Kloosterman sums. The Salié evaluation reuses the Gauss-sum sign theorem from this unit directly.
Finite fields $F_{q}$ , structure and squares 21.02.01. Every sum here lives over a finite field, and the additive- and multiplicative-character orthogonality relations that drive every proof are the dual structure of the additive group $(F_{q}, +)$ and the cyclic multiplicative group $F_{q}^{\times}$ . The trace map $Tr_{F_{q} / F_{p}}$ defining the standard additive character is the field-theoretic input from this prerequisite.
$ℓ$ -adic Galois representations 21.05.01. The Frobenius eigenvalues $α, β$ of the Kloosterman sum are the eigenvalues of Frobenius on the first $ℓ$ -adic cohomology of the Artin-Schreier curve, i.e. on a two-dimensional Galois representation. The Weil bound $∣ α ∣ = ∣ β ∣ = p$ is the purity (weight-one) statement for this representation, and the Kloosterman sheaf is the geometric incarnation studied by Katz.
Dirichlet $L$ -functions and characters [21.13.04 / 21.03.02]. Gauss sums of Dirichlet characters are the root numbers in the functional equations of Dirichlet $L$ -functions, and Kloosterman sums are the geometric side of the Kuznetsov and Petersson trace formulas, linking the analytic theory of automorphic $L$ -functions to these exponential sums. The Linnik-Selberg conjecture on sums of Kloosterman sums is a statement about cancellation beyond the Weil bound.
The large sieve and bilinear forms 21.14.04. The large-sieve inequality and its bilinear-form refinements use the Weil bound to control the off-diagonal contributions in averages of $∣ S (m, n; c) ∣$ , and Kloosterman-sum cancellation feeds directly into subconvexity bounds and the spacing of zeros of $L$ -functions.

Historical & philosophical context Master

Gauss introduced the quadratic Gauss sum in the 1801 Disquisitiones Arithmeticae and determined its sign in 1805 after four years of effort ^{[Weil 1948]}. The general multiplicative-character Gauss sum and the Jacobi sum were developed by Gauss, Jacobi, and Eisenstein in the 1830s and 1840s in connection with higher reciprocity laws and the representation of primes by quadratic and higher forms; Jacobi's 1837 computation of the number of representations of an integer as a sum of squares is the classical application of Jacobi sums. The decisive modern reframing is due to André Weil, whose 1949 Bulletin of the American Mathematical Society paper "Numbers of solutions of equations in finite fields" ^{[Weil 1949]} computed the number of points on diagonal hypersurfaces over $F_{q}$ in terms of Jacobi sums and, observing that the resulting eigenvalues all have absolute value a power of $q$ , formulated the Weil conjectures — the existence of a cohomology theory whose Lefschetz fixed-point formula would produce these counts.

Kloosterman introduced the sums bearing his name in his 1927 Acta Mathematica paper ^{[Kloosterman 1927]} as a refinement of the Hardy-Littlewood circle method, needed to handle the representation of integers by quaternary quadratic forms $a x^{2} + b y^{2} + c z^{2} + d t^{2}$ where the naive major-arc analysis fails; he proved the first non-degenerate bound $S (m, n; p) = O (p^{3/4})$ . Salié in 1932 ^{[Salié 1932]} found that the quadratic-character twist of the Kloosterman sum could be evaluated in closed form, the first indication that not all Kloosterman-type sums are equally deep. The sharp bound $∣ S (m, n; p) ∣ \leq 2 p$ was proved by Weil in 1948 ^{[Weil 1948]} as a corollary of his proof of the Riemann hypothesis for curves over finite fields, using the Artin-Schreier curve $y^{p} - y = m x + n / x$ . Stepanov 1969 ^{[Stepanov 1969]} gave an elementary proof of the curve bound via auxiliary polynomials, later systematised by Bombieri and Schmidt. The Weil conjectures were proved in full by Grothendieck's school, with the Riemann hypothesis component settled by Deligne 1974 ^{[Deligne 1974]}, giving square-root cancellation for complete exponential sums in any number of variables; Katz 1988 then computed the monodromy of the Kloosterman sheaf, establishing the Sato-Tate equidistribution of the angles.

Bibliography Master

@book{Weil1948Courbes,
  author    = {Weil, Andr{\'e}},
  title     = {Sur les courbes alg{\'e}briques et les vari{\'e}t{\'e}s qui s'en d{\'e}duisent},
  series    = {Actualit{\'e}s Scientifiques et Industrielles},
  number    = {1041},
  publisher = {Hermann},
  address   = {Paris},
  year      = {1948}
}

@article{Weil1949Numbers,
  author  = {Weil, Andr{\'e}},
  title   = {Numbers of solutions of equations in finite fields},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {55},
  year    = {1949},
  pages   = {497--508}
}

@article{Kloosterman1927,
  author  = {Kloosterman, Hendrik Douwe},
  title   = {On the representation of numbers in the form $ax^2 + by^2 + cz^2 + dt^2$},
  journal = {Acta Mathematica},
  volume  = {49},
  year    = {1927},
  pages   = {407--464}
}

@article{Salie1932,
  author  = {Sali{\'e}, Hans},
  title   = {{\"U}ber die Kloostermanschen Summen $S(u,v;q)$},
  journal = {Mathematische Zeitschrift},
  volume  = {36},
  year    = {1932},
  pages   = {263--278}
}

@article{Stepanov1969,
  author  = {Stepanov, Sergei A.},
  title   = {On the number of points of a hyperelliptic curve over a prime field},
  journal = {Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya},
  volume  = {33},
  year    = {1969},
  pages   = {1171--1181}
}

@article{Bombieri1966,
  author  = {Bombieri, Enrico},
  title   = {On exponential sums in finite fields},
  journal = {American Journal of Mathematics},
  volume  = {88},
  year    = {1966},
  pages   = {71--105}
}

@article{Deligne1974WeilI,
  author  = {Deligne, Pierre},
  title   = {La conjecture de Weil. I},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {43},
  year    = {1974},
  pages   = {273--307}
}

@book{Katz1988GaussKloosterman,
  author    = {Katz, Nicholas M.},
  title     = {Gauss Sums, Kloosterman Sums, and Monodromy Groups},
  series    = {Annals of Mathematics Studies},
  volume    = {116},
  publisher = {Princeton University Press},
  year      = {1988}
}

@book{IwaniecKowalski2004,
  author    = {Iwaniec, Henryk and Kowalski, Emmanuel},
  title     = {Analytic Number Theory},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {53},
  publisher = {American Mathematical Society},
  year      = {2004}
}

@book{IrelandRosen1990NT,
  author    = {Ireland, Kenneth and Rosen, Michael},
  title     = {A Classical Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {84},
  edition   = {2},
  year      = {1990}
}

@book{LidlNiederreiter1997,
  author    = {Lidl, Rudolf and Niederreiter, Harald},
  title     = {Finite Fields},
  series    = {Encyclopedia of Mathematics and its Applications},
  volume    = {20},
  edition   = {2},
  publisher = {Cambridge University Press},
  year      = {1997}
}

Prerequisites

21.02.02
21.01.06
21.02.01

Tier anchors

beginner: Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium Publications 53) Ch. 11 opening; Ireland-Rosen 1990 *A Classical Introduction to Modern Number Theory* (GTM 84) Ch. 8 §1-§3 informal motivation; Davenport 2008 *The Higher Arithmetic* (8th ed., Cambridge) on character sums
intermediate: Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 11 §§11.1-11.7 (Gauss, Jacobi, Kloosterman, Salié sums and the Weil bound); Ireland-Rosen 1990 (GTM 84) Ch. 8 (Gauss and Jacobi sums) and Ch. 11 (Weil's work); Lidl-Niederreiter 1997 *Finite Fields* (Encyclopedia of Mathematics 20, 2nd ed.) Ch. 5
master: Weil 1948 *Sur les courbes algébriques et les variétés qui s'en déduisent* (Hermann, Paris — the Riemann hypothesis for curves over finite fields, yielding the bound $|S(m,n;p)| \le 2\sqrt p$); Weil 1949 *Bull. Amer. Math. Soc.* 55, 497-508 (Number of solutions of equations in finite fields — the Weil conjectures and the Jacobi-sum point counts); Kloosterman 1927 *Acta Math.* 49, 407-464 (the sums $S(m,n;c)$ and the circle-method application to $ax^2+by^2+cz^2+dt^2=N$); Salié 1932 *Math. Z.* 36, 263-278 (the half-integral-weight twisted Kloosterman sums evaluable in closed form); Stepanov 1969 *Izv. Akad. Nauk SSSR* 33 (elementary polynomial proof of the Weil bound); Bombieri 1966 *Amer. J. Math.* 88 (the Stepanov method systematised); Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 11-12; Katz 1988 *Gauss Sums, Kloosterman Sums, and Monodromy Groups* (Annals of Mathematics Studies 116); Deligne 1974 *Publ. Math. IHÉS* 43 (La conjecture de Weil I — the general Weil bound via $\ell$-adic cohomology)

References

Weil, A. — Sur les courbes algébriques et les variétés qui s'en déduisent · Actualités Scientifiques et Industrielles 1041, Hermann, Paris 1948. Proves the Riemann hypothesis for curves over a finite field; the case of the Artin-Schreier curve $y^p - y = ax + b/x$ yields the Kloosterman-sum bound $|S(m,n;p)| \le 2\sqrt p$, and the case of the Fermat-type curves yields the absolute value $|J(\chi,\lambda)| = \sqrt p$ of Jacobi sums.
Weil, A. — Numbers of solutions of equations in finite fields · Bulletin of the American Mathematical Society 55 (1949), 497-508. States the Weil conjectures and computes the number of points on diagonal hypersurfaces $\sum a_i x_i^{d_i} = b$ over $\mathbb{F}_q$ in terms of Jacobi sums, exhibiting the eigenvalues of Frobenius as products of Gauss sums.
Kloosterman, H. D. — On the representation of numbers in the form $ax^2 + by^2 + cz^2 + dt^2$ · Acta Mathematica 49 (1927), 407-464. Introduces the sums $S(m,n;c) = \sum_{x} e((mx + n\bar x)/c)$ in a refinement of the Hardy-Littlewood circle method for quaternary quadratic forms, and proves the first non-degenerate bound $S(m,n;p) = O(p^{3/4})$.
Salié, H. — Über die Kloostermanschen Summen $S(u,v;q)$ · Mathematische Zeitschrift 36 (1932), 263-278. Evaluates in closed form the twisted Kloosterman sums attached to the quadratic character (the Salié sums), expressing them as a product of a quadratic Gauss sum and a single cosine, so that $|T(m,n;p)| \le 2\sqrt p$ follows by an elementary square-root extraction rather than from the Riemann hypothesis for curves.
Stepanov, S. A. — On the number of points of a hyperelliptic curve over a prime field · Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 33 (1969), 1171-1181. Gives an elementary proof of the Weil bound for the relevant curves using auxiliary polynomials of controlled degree with high-order vanishing, bypassing algebraic geometry; systematised by Bombieri 1966/1973 and Schmidt 1976.
Deligne, P. — La conjecture de Weil. I · Publications Mathématiques de l'IHÉS 43 (1974), 273-307. Proves the Riemann hypothesis component of the Weil conjectures for smooth projective varieties over finite fields via $\ell$-adic cohomology and the weight formalism, giving the general square-root cancellation bound for complete exponential sums of which the Weil bound for Kloosterman sums is the curve case.
Iwaniec, H. & Kowalski, E. — Analytic Number Theory · American Mathematical Society Colloquium Publications 53 (2004). Chapters 11-12 develop Gauss, Jacobi, Kloosterman, and Salié sums, the Weil bound, and the analytic applications (Linnik-Selberg, Kuznetsov, sums of Kloosterman sums); the modern reference for the analytic theory of complete exponential sums.
Katz, N. M. — Gauss Sums, Kloosterman Sums, and Monodromy Groups · Annals of Mathematics Studies 116, Princeton University Press 1988. Identifies the Kloosterman sums as traces of Frobenius on the Kloosterman sheaf and computes the geometric monodromy group, explaining the Sato-Tate vertical distribution of normalised Kloosterman angles.

Estimated time

beginner: 18m
intermediate: 42m
master: 80m