On the size of the maximum of incomplete Kloosterman sums

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n<H}t(n)\Big|. \] In this very general context one of the most important results is the P\'olya-Vinogradov bound \[ M(t)\leq \left\|K\right\|_{\infty}\log 3p. \] where $K:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this paper we provide a lower bound for incomplete Kloosterman sum, namely we prove that for any $\varepsilon>0$ there exists some $a\in\mathbb{F}_{p}^{\times}$ such that \[ M(e(\tfrac{ax+\overline{x}}{p}))\geq \Big(\frac{1-\varepsilon}{\sqrt{2}\pi}+o(1)\Big)\log\log p. \] Moreover we also provide some result on the growth of the moments of $\{M(e(\tfrac{ax+\overline{x}}{p}))\}_{a\in\mathbb{F}_{p}^{\times}}$.


Introduction
Let t : F p → C be a complex valued function on F p . A classical problem in analytic number theory is to bound the incomplete sums S(t, H) := 1 √ p 0≤n<H t(n), for any 0 ≤ H < p. In this very general context one of the most important results is the following: Theorem 1.1 (Pólya-Vinogradov bound, [Pol18], [Vin18]). For any 1 ≤ H < p one has where K : F p → C is the normalized Fourier transform of t K(y) := − 1 √ p 0≤x<p t(x)e yx p .
Notice that if K ∞ is bounded, then this bound is non-trivial as soon as H ≫ √ p log p.
The first question which arises in this setting is the following: given a function t : F p → C, is the Pólya-Vinogradov bound sharp for t? And if it is not, what is the best possible bound?

Kloosterman sums, Birch Sums and main results
The aim of this paper is to study the case of the Kloosterman sums and Birch sums. We recall here the definition of these two objects: i) Kloosterman sums. For any a, b ∈ F × p one considers t : x → e ax + bx p where x denotes the inverse of x modulo p. The complete sum over F × p of the function above Kl(a, b; p) := 1 √ p 1≤x<p e ax + bx p is called Kloosterman sum associated to a, b. The Riemann hypothesis over curves for finite fields implies |Kl(a, b; p)| ≤ 2 (Weil bound).
ii) Birch sums. For any a, b ∈ F × p one considers t : x → e ax + bx 3 p .
One defines the Birch sum associated to a, b Bi(a, b; p) := 1 √ p 1≤x<p e ax + bx 3 p .
Also in this case an application of the Riemann hypothesis over curves for finite field leads to the bound |Bi(a, b; p)| ≤ 2 (Weil bound).
It is known that M (e( ax+x p )) and M (e( ax+x 3 p )) can be arbitrarily large when a varies over We will prove the following lower bounds: Theorem 1.2. Let 0 < ε < 1. For all p, there exists S p ⊂ F × p such that i) for any a ∈ S p one has M (e( ax+x p )) ≥ 1 − ε √ 2π + o(1) log log p, ii) |S p | ≫ ε p 1− log(4) (log p) ε .
The same is true if one replaces M (e( ax+x p )) by M (e( ax+x 3 p )). Theorem 1.3. Let 0 < ε < 1 and fix m ≥ 1. For all p such that p ∤ m, there exists The same is true if one replaces M (e( ax+bx p )) by M (e( ax+bx 3 p )) and H m,p by C m,p .
The proofs of these two Theorems rely on the fact that we can control simultaneously the sign and the size of ∽ (log p) 1−ε Kloosterman (or Birch) sums. Indeed we will prove Proposition 1.4. For all p there exists S p ⊂ F × p such that for any a ∈ S p Kl(an, 1; p) ≥ √ 2, for any 1 ≤ n ≤ (log p) 1−ε odd, and Kl(an, 1; p) ≤ − √ 2, for any −(log p) 1−ε ≤ n ≤ −1 odd. Moreover |S p | ≫ ε p 1− log(4) (log p) ε . The same is true if we replace Kl by Bi.
In the second part of the paper, we focus our attention on the growth of the 2k-th moments of {M (e( ax+x p ))} a∈F × p and {M (e( ax+x 3 p ))} a∈F × p when p → ∞, getting Theorem 1.5. There exist two absolute positive constants C > 1 and c < 1 such that for any fixed k ≥ 1 and p → ∞ one has and for any fixed m ∈ Z \ {0} and p → ∞ Theorem 1.6. There exist two absolute constants C > 1 and c < 1 such that for any fixed k ≥ 1 and p → ∞ one has and for any fixed m ∈ Z \ {0} and p → ∞ where P (k) := exp(4k log log k + k log log log k + o(k)).
From this we get the following Corollary 1.7. There exist two absolute constants B, b > 0 such that for A → ∞ one has

Remarks and related works
i) The upper bound in Theorem 1.5 can be improved conditionally on Conjecture 1.8 (Short sums conjecture for Kloosterman sums). There exists an ε > 0 such that uniformly for any 1 < N < p, p 1/2−ε/2 < H < p 1/2+ε/2 and a ∈ F × p .
Indeed, assuming this conjecture we will prove that Notice that Conjecture 1.8 is a (much) weaker form of Hooley's R * -assumption ([Hoo78, page 44]). In the case of the moments of maximum of incomplete Birch sums we get a better upper bound since the analogue of the (1) is known to be true for the function x → e ax+bx 3 p (Weyl's inequality).
Assuming Conjecture 1.8, we can prove the same in the case of Kloosterman sums thanks to the fact that for any (a, b) ∈ H m,p Kl(a, b; p) = Kl(m, 1; p).
iii) Lamzouri in [Lam18] has proved that there exist some (computable) constants C 0 , C 1 and δ such that for any 1 ≪ A ≤ 2 π log log p − 2 log log log p one has He obtains the same result also for incomplete Kloosterman sums. For the family {M (e( ax+x 3 p ))} a∈F × p , he also proved that Also in this case the difference between the incomplete Kloosterman sums and incomplete Birch sums depends on the cancellation of the short sums of Kloosterman sums (Conjecture 1.8). The proof of the lower bound in (2) implies that for at least where t a = e( ax+x 3 p ) or t a = e( ax+x p ).
iv) One should compare our result with the case of incomplete character sums. Paley proved that the Pólya-Vinogradov bound is close to be sharp in this case: indeed in [Pal32] is shown that there exist infinitely many primes p such that where · p is the Legendre symbol modulo p. Similar results were achieved for nontrivial characters of any order by Granville and Soundararajan in [GS07], and by Goldmakher and Lamzouri in [GL12] and [GL14]. On the other hand Montgomery and Vaughan have shown under G.R.H. that for any χ ( [MV77]), which is the best possible bound up to evaluation of the constant.
Acknowledgment. I am most thankful to my advisor, Emmanuel Kowalski, for suggesting this problem and for his guidance during these years. I also would like to thank Youness Lamzouri for informing me about his work on sum of incomplete Birch sums.

Notation and statement of the main results
In this section we recall some notion of the formalism of trace functions and state the general version of our main results. For a general introduction on this subject we refer to [FKM14]. Basic statements and references can also be founded in [FKM15a]. ii) The Birch sums: b → Bi(a, b; p) it can be seen as the trace function attached to the sheaf FT(L e((aT 3 )/p) ) iii) The n-th Hyper-Kloosterman sums: the map can be seen as the trace function attached to the Kloosterman sheaf Kℓ n (see [Kat88] for the definition of such sheaf and for its basic properties).
Definition 1.2. Let p, ℓ > 2 be a prime numbers with p = ℓ and let r ≥ 1 be an integer.
ii) the geometric and arithmetic monodromy groups of F satisfy G arith ii) the projective automorphism group Definition 1.3. Let p, ℓ > 2 be a prime numbers and let r ≥ 1 be an integer. A r-family (F a ) a∈F × p is r-acceptable if the following conditions are satisfied: i) for any a ∈ F × p , F a is an irreducible middle-extension ℓ-adic Fourier sheaf on A 1 Fp pointwise pure of weight 0. We denote by t a the trace function attached to F a .
ii) The ℓ-adic Fourier transform FT(F 1 ) is an r-bountiful sheaf, for any a ∈ F × p , where K a (·) denote the trace functions attached to FT(F a ).
for any p prime and a ∈ F × p . We call the smallest C with this property the conductor of the family and we denote it by C F . Definition 1.5. Let F be a r-coherent family and for any A > 0 we define Example 1.1. The following families are 2-coherent: i) The family of Artin-Schreier sheaves so we can take τ y := 1 y 0 1 .
ii) The family of Artin-Schreier sheaves It is enough to argue as above and to observe that Then Theorem 1.2 and 1.3 are consequences of the following Similarly, Theorem 1.5 and 1.6 are consequence of: There exist two positive constant C > 1 and c < 1 depending only on c F such that for any fixed k ≥ 1 If moreover one has that there exists an ε > 0 such that uniformly for any 1 < N < p, p 1/2−ε/2 < H < p 1/2+ε/2 and a ∈ F × p , then for any fixed k ≥ 1 one has We then get the following Corollary 1.11. Same notation as in Theorem 1.10. Then: 2 Proof of Theorem 1.9 First step: Fourier expansion and Féjer Kernel The first step for both Theorems is to get a quantitative version of the Fourier expansion for 1 √ p x≤αp t(x): Lemma 2.1. Let t : F p → C be a complex valued function on F p , then for any for any 0 < α < 1 we have for any 1 ≤ N ≤ p, where the implied constant is absolute.
Proof. We use the same strategy used in [Pol18]. Let us introduce the function Then the Fourier series of Φ is Observe that for any N > 1 one has On the other hand we have .
Then one has and similarly Now we use the same strategy of [Pal32] introducing the Fejér's kernel: Lemma 2.2. For any t : F p → C one has Proof. The quantitative version of the Fourier transform leads to at this point we extend the outer sum to all values modulo p using the Fejér's kernel: for any 1 < N < p we have where φ(a) : is the Féjer Kernel, and (1 − e(−αn))e(−ϑn).
On the other hand the triangular inequality leads to So we obtain the bound On the other hand using the fact we conclude the proof.
To conclude, it is enough to prove the following Proposition 2.3. Same assumption as in Theorem 1.9. Let 0 < ε < 1. Then for all p there exists S p ⊂ F × p such that for any a ∈ S p K 1,p (τ n · a) ≥ √ 2, for any 1 ≤ n ≤ (log p) 1−ε odd, and: for any −(log p) 1−ε ≤ n ≤ −1 odd. Moreover |S p | ≫ ε,c F p 1− log(4) (log p) ε . Assuming this Proposition, which we prove in the next section, let us prove Theorem 1.9. We have that for any a ∈ S p , where in the second step uses the fact that K a,p (n) = K 1,p (τ n · a) (the family is 2-bountiful).

Proof of Lemma 2.3 via Chebyshev Polynomials
From now on p is a fixed prime number. We consider an irreducible 2-bountiful sheaf K on Fp and we will denote the trace function attached to it by K(·). The 2-bountiful condition on the sheaf K implies that for any a ∈ F p , one has K(a) = 2 cos(θ(a)).
with θ(a) ∈ [0, π]. We call θ(a) the angle associated to K(a). We recall that there exist polynomials U n for n ≥ 0 such that U n (2 cos θ) = sin((n + 1)θ) sin θ , for all θ ∈ [0, π]. In terms of Representation Theory, these are related to the characters of the symmetric power of the standard representation of SU 2 . In particular by Peter-Weyl Theorem, these form an orthonormal basis of L 2 ([0, π], µ ST ). Note that we can see U n (K(·)) as the trace function attached to the sheaf Sym n (K). Moreover we call trigonometric polynomial of degree s ≥ 0 any Y ∈ L 2 ([0, π], µ ST ) written in the form with y(s) = 0. Let us start by proving some property of the sheaf Sym n (K): Lemma 2.4. Let K as above. For any n > 0: i) The geometric monodromy group of Sym n (K) is given by ii) The projective automorphism group Aut 0 (Sym n (K)) := {γ ∈ PGL 2 (F p ) :γ * Sym n (K) ∼ = Sym n (K) ⊗ L for some rank 1 sheaf L} is trivial.
iii) The conductor of Sym n (K) is bounded by c(Sym n (K)) ≤ n · c(K).
Proof. Let us start with part (i): by the definition of the geometric monodromy one has that G geom Sym n (K) = Sym n (G geom K ). Then the result follows because G geom K = SU 2 by hypothesis. Let us prove now part (ii). Let γ ∈ PGL 2 (F p ). First observe that t Sym n (K) (x) = sin((n + 1)θ(x)) sin(θ(x)) , t γ * Sym n (K) (x) = sin((n + 1)θ(γ · x)) sin(θ(γ · x)) , where t K (x) = 2 cos θ(x). Thanks to the fact that K is a bountiful sheaf we know that the r → ∞ (Goursat-Kolchin-Ribet criterion). By contradiction, assume that γ * Sym n (K) ∼ = Sym n (K) ⊗ L for some rank 1 sheaf. We may assume that L is of weights 0. Let U be a dense open set where γ * Sym n (K), Sym n (K) and L are lisse. Using the equidistribution we can find x ∈ U such that On the other hand in U one would have and this is absurd. Part (iii) is just a consequence of Deligne's Equidistribution Theorem (see for example [Kat88, Paragraph 3.6]).
Lemma 2.5. Let (Y i ) n i=0 be a family of trigonometric polynomials as above such that for where y = max i,j |y i (j)| and the constant C is absolute.
Proof. To prove the lemma it is enough to bound The result then follows from the fact that Rank Sym ni (K)) = n i + 1 ≤ 2d, c(Sym ni (K)) ≤ n i c(K) ≤ dc(K), because n i ≤ d for all i by assumption.

Proof of Proposition 2.3
We can now prove Proposition 2.3. Let z ∈ N be an odd positive number and let γ ∈ N, we denote by θ := (θ 2j−z ) z j=0 ∈ [0, π] z+1 and by χ 1 γ (·) (resp. χ − 1 γ (·)) the characteristic function of [0, π 2 − π γ ] (resp.[ π 2 + π γ , π]). To prove Proposition 2.3 we start approximating the function using Chebyshev polynomials. We use the same method adopted in [KLSW10, Section 3]: for any z, we find an integer L ≡ −1 mod 2γ and two families of trigonometric polynomials {α i }, and {β i } such that if we define the following inequality holds for any θ ∈ [0, π] z+1 . Moreover we will prove Lemma 2.6. With the notation as above, we have: i) There exist two constant L 0 ≥ 1 and c > 0 depending only on γ, such that the contribution ∆ of the constant term in the Chebyshev expansions of A L θ π − B L θ π satisfies: if L is the smallest integer such that L ≡ −1 mod 2γ satisfying L ≥ max(cz, L 0 ).
ii) All coefficients in the Chebyshev expansion of the factors in A L θ π and the terms in B L θ π are bounded by 1.
iii) The degrees, in terms of the Chebyshev expansion, of the factors of A L θ π and B L θ π are ≤ 2L.
Once we have this Lemma we can easily get Proposition 2.3. Fix γ = 1 4 in Lemma 2.6 and denote S p the set of a ∈ F × q which satisfy the property in the Proposition 2.3. Let L be as in part (i) of Lemma 2.6, then we have where in the second step we are using Lemma 2.5, notice that i) The condition τ i = τ j if i = j is satisfied by definition of acceptable family.
ii) thanks to part (ii) of Lemma 2.6 we have that y in Lemma 2.5 is equal to 1.
Proof of Lemma 2.6. The main references for this proof are [KLSW10, Lemma 3.2] and [BMV01]. We define We can rewrite the above trigonometric polynomials as Remember that the n-th coefficient in the Chebychev expansion of α L,± and β L,± are given by π 0 α L,± ( θ π )U n (θ)dµ st , π 0 β L,± ( θ π )U n (θ)dµ st , then part (iii) immediately follows because the above integrals vanishes if n > 2L. Moreover in [BMV01,Lemma 5] it is shown that 0 ≤ α L,± (x) ≤ 1 for x ∈ [0, 1] and the same holds for the |β L,± |s by definition. Using Cauchy-Schwarz inequality we get the same argument can be used for β L,± and this proof part (ii). It remains to prove only part (i), as we have just observed for any trigonometric polynomial Y the constant term of its Chebyshev expansion is given by so we have that ∆ in part (i) is given by Using the definition of β L,± we get π 0 β L,± ( θ π )dµ st = 1 L + 1 , so we can write ∆ as When L → ∞, α L,± → χ ± 1 γ in L 2 ([0, 1]), moreover from [BMV01, (2.6)] one has |χ ± 1 γ − α L,± | ≤ |β L,± | 0 ≤ x ≤ 1 and from the Fourier expansion of β L,± (x) we have This implies that there exist L 0 , such that the integral in the left hand side of the equation above is ≥ 1 2 − 1 γ so we get: If we assume 2L + 2 ≥ 6z 1 2 − 1 γ −2 we get: as we wanted.
By the triangular inequality one gets where in the first inequality we use the fact that K a,p (0) ≤ c(F a,p ) ≤ c F by hypothesis. To conclude the proof of the Lemma it is enough to provide a bound for the first term in the right hand side. Extending the 2k-power we get An application of [FKM15b, Corollary 1.7] implies a∈F × p K 1,p (τ n1 · a) · ... · K 1,p (τ n k · a)K 1,p (τ l1 · a) · ... · K 1,p (τ l k · a) − m(n, l)p ≤ δ 2k On the other hand |c n | ≤ 2 min 1 n , β − α π ≤ 2 n , hence we get B ≤ √ pδ 2k 2 (log p) 2k for some δ 2 > 0 depending only on c F . To bound A we can proceed as follows: first observe that if m(n, l) = 0 then there exists a constant dependent only on c F , let's say γ 1 , such that m(n, l) ≤ γ 2k 1 (again the reference here is [FKM15b, Corollary 1.7]). Thus A ≤ γ 2k 1 p n (n1,...,n 2k )∈m(n) c n1 · · · c n 2k , where m(n) := {(n 1 , ..., n 2k ) : n 1 · · · n 2k = n any n i appears an even number of times}.