Nearly sharp Lang–Weil bounds for a hypersurface

Abstract We improve to nearly optimal the known asymptotic and explicit bounds for the number of 
$\mathbb {F}_q$
 -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.


Introduction
Let n ≥ 2, d ≥ 1, and let F q be a finite field.Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Lang and Weil [4] have established the bound where C d depends only on d and n.We summarize the smallest known possible values of C d available in the literature.
e) The author's Theorem 8 in the preprint [7] implies that for every ε > 0, ε ′ > 0, we have as long as q ≫ 1 (again with an explicit condition on q).
In this note we tighten the known asymptotic and explicit bounds for |X(F q )| when q is large relative to d.
We first look at upper bounds.
Theorem 1.Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Then where the implied constant depends only on d and can be computed effectively.
We can exhibit explicit bounds, as in the theorem below.
Theorem 2. Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Suppose that q > 15d 13/3 .Then Example 3 (Cylinder over a maximal curve).Let d ≥ 3 be such that d − 1 is a prime power.Let q be an odd power of (d − 1) 2 .Consider the curve Fq .It is known (see, for example, [8]) that #C(F q ) = q + (d − 1)(d − 2) √ q.Thus the number of Example 3 is nonsingular, its Zariski closure in P n has a large (in fact, (n − 3)-dimensional) singular locus.In general, let X ⊂ A n be a geometrically irreducible hypersurface such that #X(F q ) ≥ q n−1 + (d − 1)(d − 2)q n−3/2 − O d (q n−2 ) for large q.Theorem 6.1 in [3] implies that the Zariski closure of X in P n must have singular locus of dimension n − 3 or n − 2.
As in Theorem 4 in [6], we can exhibit a forbidden interval for |X(F q )|.Notice that X is not necessarily geometrically irreducible in the statement below.

Theorem 5. Let X ⊂ A n
Fq be a hypersurface of degree d.If Remark 6.Let us write g(d) + • • • for an effectively computable g(d) + g 1 (d), where g 1 (d) = o(g(d)) for d → ∞.Theorem 5 has content when the right-hand side of (5) exceeds the righthand side of (6), which takes place for q > 16d 4 + • • • .Thus in the presence of Theorem 2, Theorem 5 addresses the range 16d 4 + • • • < q < 15d 13/3 .Notice that in the Lang-Weil bound (1), the approximation term q n−1 dominates the error precisely when q > d 4 + • • • .This is why it is reasonable to frame the entire discussion of the Lang-Weil bound in the range q > d 4 + • • • .For example, any lower Lang-Weil bound is trivial for q below this threshold.
We improve the lower bounds for |X(F q )| as well.The proof of Theorem 4 in [6] actually gives a lower bound which is tighter for q ≫ 1 than the one stated in [6].
Fq be a geometrically irreducible hypersurface of degree d.Then where the implied constant depends only on d and can be computed explicitly.
We give a version with an explicit lower bound as well.
Theorem 8. Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Suppose that q > 15d 13/3 .Then Example 9.As in Example 3, let d ≥ 3 be such that q 0 := d − 1 is a prime power.The curve {y d−1 z + yz d−1 = x d } in P 2 over F q 0 intersects the line x = 0 at d distinct points defined over an extension F q 1 of F q 0 .Let q be an even power of q 1 .Then the affine curve C := In fact, the proofs of Theorems 1 and 7 give an algorithm that takes as input a half-integer r ≥ 0 and constants 1 C (j) and and returns as output four additional , and and Initiating the algorithm with r = 0 and the rather weak version of ( 1), we obtain (3) and (7).In turn, taking ( 3) and ( 7) as input, we obtain Corollary 10.Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Then A lower Lang-Weil bound can be useful in proving that a geometrically irreducible hypersurface X ⊂ A n Fq has an F q -rational point.It is known (see Theorem 5.4 in [2] and its proof) that if q > 1.5d 4 + • • • , then X(F q ) = ∅.Notice that the approximation term q n−1 in (9) dominates the remaining explicit terms already for q > d 4 + • • • .Based on this heuristic, we state Conjecture 11.There exists an effectively computable function g 1 (d) = o(d 4 ) as d → ∞ with the following property.Let X ⊂ A n Fq be a geometrically irreducible hypersurface of degree d.Then X(F q ) = ∅ as long as q > d 4 + g 1 (d).
In contrast to the upper bounds, all lower bounds above (including (2) and Example 9) contain a d in the coefficient of q n−2 .This discrepancy disappears if we work in projective space.
Theorem 12. Let X ⊂ P n Fq be a geometrically irreducible hypersurface of degree d.Then ) and Example 13 (Cone over a maximal curve).Let (d, q 0 ) be such that there exists a (nonsingular) maximal curve C = {f = 0} in P 2 over F q 0 of degree d.Let q be a power of q 0 and let X = {f = 0} ⊂ P n Fq be a projective cone over C. Then with ± depending on whether q is an odd or an even power of q 0 .Thus the gap between what is achieved in this example and what is established in Theorem 12 is q n−2 + O d (q n−5/2 ) in the case of the lower bound and (π 2 /6)q n−2 + O d (q n−5/2 ) in the case of the upper bound.
This paper builds upon the author's earlier work [6] and is inspired by T. Tao's discussion [9] of the Lang-Weil bound through random sampling and the idea of Cafure-Matera [2] to slice X with planes.A plane is a 2-dimensional affine linear subvariety of A n Fq .If H ⊂ A n Fq is any plane, then #(X ∩H)(F q ) is either q 2 , 0, or ≈ kq, where k is the number of geometrically irreducible F q -irreducible components of X ∩ H.For 0 ≤ k ≤ d, we exhibit a small interval The problem when it comes to the upper bound is that when k is large, planes H with #(X ∩ H)(F q ) ∈ I k contribute significantly towards the count #X(F q ).However, it turns out that the number of such H's decreases quickly as k grows.

A collection of small intervals Lemma 14 ([5], Lemma 5). Let C ⊂ A 2
Fq be a curve of degree d.Let k be the number of geometrically irreducible F q -irreducible components of C. Then It will be crucial to give a refined upper bound when k = 1.
Lemma 15.Let C ⊂ A 2 Fq be a curve of degree d.Suppose that C has exactly one geometrically irreducible F q -irreducible component.Then Proof.Let C 1 , . . ., C s be the F q -irreducible components of C. Suppose that C 1 is geometrically irreducible, but C i is not for i ≥ 2. Let e = deg(C 1 ).Note that (d, e) = (2, 1).Using the Aubry-Perret bound (2) for C 1 and Lemma 2.3 in [2] for each C i with i ≥ 2, we estimate to justify the last inequality in the chain, note that it is equivalent to and holds true because either e = d, or else d − e > 0 and we can write > 0 (using that e ≥ 1 and d ≥ 3 on the last step). Let Fq be a hypersurface of degree d.Let H ⊂ A n Fq be a plane.Then #(X ∩ H)(F q ) ∈ I k for some k ∈ {0, . . ., d} ∪ {∞}.
we use Lemma 15 and the lower bound from (2) applied to a geometrically irreducible F q -irreducible component (necessarily of degree ≤ d) of X.For 2 ≤ k ≤ d, use Lemma 14.
Alternatively, one could take b d = dq by the Schwartz-Zippel lemma.When it comes to giving an upper bound for |X(F q )|, it will be more convenient to work with J 1 := I 0 ∪ I 1 and J i := I i for i ∈ {2, . . ., d} ∪ {∞}.

Probability estimates
We spell out in detail the proof of Theorem 1; the proofs of the remaining results will then require only slight modifications.The implied constant in each O-notation is allowed to depend only on d (a priori, possibly also on n), but not on q or X.
Proof of Theorem 1. Set N := |X(F q )|.For a plane H ⊂ A n Fq chosen uniformly at random, consider #(X ∩ H)(F q ) as a random variable.Let µ and σ 2 denote its mean and variance.Lemma 10 in [6] and (1) imply For k ∈ {1, . . ., d} ∪ {∞}, denote We can assume that q is large enough so that the intervals J 1 , . . ., J d are pairwise disjoint.
Define t via (k − 1)q − O( √ q) = tσ; then Chebyshev's inequality and the variance bound (10) imply We now go back to (11) and apply the Abel summation formula: Multiply both sides by q n−2 to arrive at (3).
Going through all the explicit inequalities with a O-term, one can compute explicitly a possible value of the constant implicit in (3).In fact, since there is a choice of C d in the Lang-Weil bound that depends only on d and not on n, a second look at all the inequalities written down in the proof above reveals that the implied constant in (3) can likewise be chosen to not depend on n.
For the rest of the paper, we follow the notation and proof of Theorem 1.
Proof of Theorem 7. Say that a plane H is "bad" if #(X∩H)(F q ) ∈ I 0 and "good" otherwise.
Fq is a bad plane, then By computations similar to the ones in the proof of Theorem 1, the probability that a plane is bad is at most q −1 + O(q −3/2 ).Every good plane contributes at least a 1 to the mean.Therefore √ q − d + 1), giving (7).
Proof of Corollary 10.Modify the proof of Theorem 7, but use the upper bound for N from (3) and the lower bound for N from ( 7) respectively for the upper bound on σ 2 and the lower bound on N/q n−2 − d 2 /4.
It remains to check that the function g(d) on the right-hand side above satisfies g(d) > 7.44 for any integer d ≥ 2. On the one hand, g grows like d 1/6 so one easily exhibits a d 0 such that g(d) > 7.44 for d > d 0 .Then a simple computer calculation checks that g(d) > 7.44 for integers d ∈ {2, . . ., d 0 } as well.