Weil's Estimate Research Articles

Arithmetic progressions in certain subsets of finite fields

In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp={t2:t∈Fp} and Cp={t3:t∈Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k≥3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given byp22k+R, where|R|≤(k−24−k−22k−1)⋅p32+ck⋅p and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length ℓ in the set Sp when ℓ∈{3,4,5} and p is an odd prime number. For ℓ=4,5, our formulas are based on the number of points on certain elliptic curves, and the error term is best possible due to the Sato-Tate conjecture.

Estimates for the number of rational points on simple abelian varieties over finite fields

Let A be a simple Abelian variety of dimension g over the field $$\mathbb {F}_q$$ . The paper provides improvements on the Weil estimates for the size of $$A(\mathbb {F}_q)$$ . For an arbitrary value of q we prove $$(\lfloor (\sqrt{q}-1)^2 \rfloor + 1)^g \le \#A(\mathbb {F}_q) \le (\lceil (\sqrt{q}+1)^2 \rceil - 1)^{g}$$ holds with finitely many exceptions. We compute improved bounds for various small values of q. For instance, the Weil bounds for $$q=3,4$$ give a trivial estimate $$\#A(\mathbb {F}_q) \ge 1$$ ; we prove $$\# A(\mathbb {F}_3) \ge 1.359^g$$ and $$\# A(\mathbb {F}_4) \ge 2.275^g$$ hold with finitely many exceptions. We use these results to give some estimates for the size of the rational 2-torsion subgroup $$A(\mathbb {F}_q)[2]$$ for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.

A SIMPLE PROOF OF CHEBOTAREV’S DENSITY THEOREM OVER FINITE FIELDS

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.

On the involution fixity of exceptional groups of Lie type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

Upper bounds for some Brill–Noether loci over a finite field

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

Algebraic Cayley graphs over finite fields

A new algebraic Cayley graph is constructed using finite fields. It provides a more flexible source of expander graphs. Its connectedness, the number of connected components, and diameter bound are studied via Weil's estimate for character sums. Furthermore, we study the algorithmic problem of computing the number of connected components and establish a link to the integer factorization problem.

A twisted theorem of Chebotarev

We prove a function-field version of Chebotarev's density theorem in the framework of difference algebraic geometry by developing the notion of Galois coverings of generalized difference schemes, and using Hrushovski's twisted Lang–Weil estimate.

Existence of 2-$(v,k,1)$ Designs Admitting a Block-Transitive Group of Affine Type

In this paper we use Weil's estimate on character sums to prove that large admissible prime powers $q$ admit 2-$(q,k,1)$ designs having block-transitive automorphism groups in AGL$(1, q)$.

A twisted theorem of Chebotarev

The classical function field version of Chebotarev's Theorem follows from the Lang–Weil estimate and an ‘untwisting’ trick. We obtain an analogue in the framework of difference schemes, using Hrushovski's twisted Lang–Weil estimate. To cite this article: I. Tomašić, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On an identity associated with Weil's estimate and its applications

In this paper we study the fourth mean value of generalized quadratic Gauss sums, and get an interesting identity associated with Weil's estimate, ∑ a = 2 p − 2 ∑ b = 1 p − 1 ( a 2 − b 2 p ) ( b 2 − 1 p ) = { 10 − 2 p , p ≡ 1 (mod 4) ; 2 p − 6 , p ≡ 3 (mod 4) . By virtue of the identity, higher-order mean values of generalized quadratic Gauss sums are obtained.

A Health Plan to Reduce Poverty

Noting that the failures of the U.S. health care system are compounding the problems faced by low-income Americans, Alan Weil argues that any strategy to reduce poverty must provide access to health care for all low-income families. Although nearly all children in families with incomes under 200 percent of poverty are eligible for either Medicaid or the State Children's Health Insurance Program (SCHIP), the parents of poor children often lack health insurance. Parents who leave welfare normally get a year of coverage but then lose coverage unless their employer provides it, and many employers of low-wage workers do not offer health insurance. Similarly, parents who take low-paying jobs to avoid welfare usually have no coverage at all. This lack of coverage discourages adults from working and may also affect the health of children because adults without health insurance are less likely to take their children for preventive care. Weil proposes creating a federal earned income health credit (EIHC) and redefining the federal floor of coverage through Medicaid and SCHIP. His aim is to make health insurance affordable for low-income families and to make sure enough options are available that individuals and families can get coverage using a combination of their own, their employer's, and public resources. Weil would expand Medicaid eligibility to include all families whose income falls below the poverty line. The EIHC would be a refundable tax credit that would be available to parents during the year in advance of filing a tax return. The credit, which would be based on taxpayer earnings and family structure, would phase in as earnings increase, reach a plateau, and then phase out farther up the income scale. The credit would be larger for families with dependents. The EIHC would function seamlessly with the employee payroll withholding system. It would be available only to adults who demonstrate that they had health insurance coverage during the year and, for adults with children, only if their eligible dependent children were enrolled in either a private or public insurance program. Weil's proposal would cover individuals who receive coverage from their employer and those who do not. The proposal smooths transitions from public to private coverage, and it anticipates a substantial role for states. Weil estimates that his policy would cost about $45 billion a year.

Character sums over norm groups

In this paper, we give estimates of character sums of Weil type. They are associated with polynomials and rational functions with coefficients in a quadratic extension k 2 of a finite field k of q elements, additive and multiplicative characters of k 2 , and summed over the kernel of the norm map from k 2 × to k × . The bounds are of the magnitude O ( q ) with the implied constant depending on the degree of the polynomial and the total degree of the support of the rational function, similar to Weil's estimates over k.

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields \(F_Q \). The Weil estimate of such sums is well known: \(|S(f)| \leqslant ({\text{deg }}f - 1)\sqrt Q \), where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, \((Q,2) = 2\), for which \(|S(f)|\) attains the largest possible value and, in particular, \(|S(f)| = ({\text{deg }}f - 1)\sqrt Q \).

On Two Exponential Sums and Their Applications

We give estimates of two exponential sums over finite fields for which Weil's estimates fail. Using our estimates and Cohen's sieve method, we prove the conjecture of Hansen and Mullen for the second coefficient in characteristic two when the degree ≥7.

ON THE DISTRIBUTION IN ARITHMETIC PROGRESSIONS OF REDUCIBLE QUADRATIC POLYNOMIALS

By using Weil's estimate for Kloosterman sums, we obtain a result on the distribution of the sequence n(n+2), beyond the classical level, when a bilinear form with support over pairs of prime moduli is considered. We also obtain an analogous result in the case of a trilinear form, but only by using the recent results of Deshouillers-Iwaniec for sums of Kloosterman sums. Furthermore, the method is extended to general reducible quadratic polynomials.

An improved Bombieri-Weil bound and applications to coding theory

Recently we obtained a sharpening of the Carlitz Uchiyama bound in characteristic two using Serre's improvement on Weil's estimate of the number of rational points on an algebraic curve over a finite field. In this paper we derive an improvement on a theorem of Bombieri and Weil. This has important applications in the theory of codes; for instance we obtain immediate results concerning the minimum distance of the dual of an arbitrary geometric Goppa code and the minimum distance of reversible BCH codes of length 2 m + 1.

Estimates of the eigenvalues of Hecke operators in the space of cusp forms of weight O

From the summation formulas for general Kloosterman sums and from Weil's estimates, one derives nontrivial estimates for the eigenvalues of Hecke operators, acting in the Hilbert space of parabolic forms of weight 0 relative to the group Γo(N).

Estimates for exponential sums

If f is a polynomial over Z of degree n + 1 n + 1 with n ⩾ 1 n \geqslant 1 , then for each integer q ⩾ 1 , | Σ 1 ⩽ x ⩽ q exp ⁡ ( 2 π i f ( x ) / q ) | ⩽ q 1 / 2 ( D , q ) d n ( q ) q \geqslant 1,|{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)| \leqslant {q^{1/2}}(D,q){d_n}(q) , provided the discriminant D of the derivative of f does not vanish identically, where d n ( q ) {d_n}(q) is the number of representations of q as a product of n factors.

Algebraic actions of discrete groups: the $p$-adic method

We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on $p$-adic analysis, the second makes use of isoperimetric inequalities and Lang–Weil estimates. For instance, we show that, if $\mathsf{SL}_n(\mathbf{Z})$ acts faithfully on a complex quasi-projective variety $X$ by birational transformations, then $\mathrm{dim}(X) \geqslant n-1$ and $X$ is rational if $\mathrm{dim}(X) = n-1$.