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Math 259: Introduction to Analytic Number Theory An application of Kloosterman sums
As promised, here is the analytic lemma from [Merel 1996]. The algebraic exponential sum that arises naturally here also arises in our investigation of the coefficients of modular forms.
Fix a prime p and a nonzero c mod p. [More generally we might ask the same question for any integer N and c ∈ (Z/N )∗; see Exercise 1 below.] Let I, J ⊂ (Z/p) be intervals of size A, B < p. How many solutions (x, y) ∈ I × J are there to xy ≡ c mod p?
As usual we cannot reasonably hope for a meaningful exact answer, but on probabilistic grounds we expect the number to be roughly AB/p, and can ana- lytically bound the difference between this and the actual number:
Lemma 1. The number M of solutions (x, y) ∈ I × J of xy ≡ c mod p is AB/p + O(p^1 /^2 log^2 p).
Proof : Let χ, ψ : (Z/p)→C be the characteristic functions of I, J. Then the number of solutions to our equation is
M =
n∈(Z/p)∗
χ(n)ψ(cn−^1 ).
As it stands, this “formula” for M is just restating the problem. But we may expand χ, ψ in discrete Fourier series:
χ(x) =
a mod p
χ ˆ(a)ep(ax), ψ(x) =
b mod p
ψ^ ˆ(b)ep(bx).
[NB for t ∈ (Z/p) the notation ep(t) now means e(t/p) = e^2 πit/p, not e(pt) as before.] So, we have
M =
x mod p
a,b mod p
χ ˆ(a) ψˆ(b)ep(ax + bcx−^1 ) =
a,b mod p
χ ˆ(a) ψˆ(b)Kp(a, bc), (1)
where for a, b mod p the Kloosterman sum Kp(a, b) is defined by
Kp(a, b) :=
n∈(Z/p)∗
ep(an + bn−^1 ).
Now clearly Kp(0, 0) = p − 1, and almost as clearly Kp(0, b) = Kp(a, 0) = −1 if a, b 6 = 0. The interesting case is a, b ∈ (Z/p)∗. We now encounter yet another algebraic result that we must cite without proof, again due to Weil [1948]: for a, b nonzero,
|Kp(a, b)| < 2
p. (2)
[This comes from an interpretation of Kp(a, b) as λ + λ¯ where λ is an eigenvalue of Frobenius for the “Artin-Schreier curve” Y p^ − Y = aX + b/X, though even
the connection with that curve is nontrivial — see [1948] again, which as usual generalizes to finite fields which need not have prime order.] Putting this into (1) we find
|M − χˆ(0) ψˆ(0)(p − 1)| < 2
p
a mod p
| χˆ(a)| ·
b mod p
| ψˆ(b)|.
But ˆχ(0) = A/p and ψˆ(0) = B/p. For nonzero a, b mod p we obtain ˆχ(a), ψˆ(b) as sums of geometric series and find (as in Polya-Vinogradov)
p | χˆ(a)| {a/p}−^1 , p | ψˆ(b)| {b/p}−^1.
Thus
a |^ χˆ(a)|,^
b |^ ψˆ(b)| ^ log^ p, and Lemma 1 is proved.^
Corollary. (“Lemme 5” of [Merel 1996]) If AB is a sufficiently high multiple of p^3 /^2 log^2 p then there are x ∈ I, y ∈ J such that xy ≡ c mod p.
For instance it is enough for A, B to both be sufficiently high multiples of p^3 /^4 log p. Presumably p^1 /2+^ suffices, but as far as I know even pθ^ for any θ < 3 /4 is a difficult problem. We can, however, remove the log factors from the Corollary:
Lemma 2. Suppose I, J ⊂ (Z/p) are intervals of sizes A, B with AB ≥ 8 p^5 /^2 /(p − 1). Then there are x ∈ I, y ∈ J such that xy ≡ c mod p.
Proof : The idea is to replace χ, ψ by functions f, g : (Z/p)→[0, 1] supported on I, J whose discrete Fourier coefficients decay more rapidly than p/{ ap }, and
sum to O(1) instead of O(log p). This will yield an estimate on
M ′^ :=
n∈(Z/p)∗
f (n)g(cn−^1 )
instead of M , but if there are no solutions (x, y) ∈ I × J of xy ≡ c mod p then M ′^ vanishes as well as M and a contradiction would arise just the same.
Let χ 0 be the characteristic function of an interval of size A′^ = dA/ 2 e, and let f 0 be the convolution χ 0 ∗χ 0. Then f 0 is a function from (Z/p) to [0, A′], supported on an interval of size ≤ A centered at the origin, and with nonnegative discrete Fourier coefficients fˆ 0 (a). Thus
∑
a mod p
| fˆ 0 (a)| =
a mod p
f^ ˆ 0 (a) = f (0) = A′.
Moreover
x mod p f^0 (x) =^ A
′ (^2). Let f be a translate of f 0 /A ′ (^) supported on I.
Then
a mod p |^ fˆ (a)| = 1 and ∑ x mod p f^0 (x) =^ A
′. Define ψ 0 , g 0 , g similarly.
Arguing as before, we find that
|M ′^ −
p − 1 p^2
A′B′| < 2
p.
Thus if M ′^ = 0 then A′B′^ < 2 p^5 /^2 /(p − 1) and AB ≤ 4 A′B′^ < 8 p^5 /^2 /(p − 1), Q.E.D.
[Katz 1988] Katz, N.: Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ 1988 (#116 in The Annals of Math. Studies). [QA246.8.G38 K37]
[Merel 1996] Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437–449.
[Weil 1948] Weil, A.: On some exponential sums. Item 1948c (pages 386–389) in his Collected Papers I. [O 9.79.1 (I) / QA3.W43].
