New curves of Kummer type with many rational points over finite fields

Discussion of theory and applications of algebraic curves over finite fields with many rational points.

For each integer $k \\in [0,9]$, we count the number of plane cubic curves\ndefined over a finite field $\\mathbb{F}_q$ that do not share a common component\nand intersect in exactly $k\\ \\mathbb{F}_q$-rational points. We set this up as a\nproblem about a weight enumerator of a certain projective Reed-Muller code. The\nmain inputs to the proof include counting pairs of cubic curves that do share a\ncommon component, counting configurations of points that fail to impose\nindependent conditions on cubics, and a variation of the MacWilliams theorem\nfrom coding theory.\n

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

We estimate the average cardinality $\mathcal{V}(\mathcal{A})$ of the value set of a general family $\mathcal{A}$ of monic univariate polynomials of degree $d$ with coefficients in the finite field $\mathbb{F}_{\hskip-0.7mm q}$. We establish conditions on the family $\mathcal{A}$ under which $\mathcal{V}(\mathcal{A})=\mu_d\,q+\mathcal{O}(q^{1/2})$, where $\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any restriction on the characteristic of $\mathbb{F}_{\hskip-0.7mm q}$ and provides an explicit expression for the constant underlying the $\mathcal{O}$--notation in terms of $d$. We reduce the question to estimating the number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over $\mathbb{F}_{\hskip-0.7mm q}$. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points.

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type \({^2A_2,\,^2B_2}\) and \({^2G_2}\) defined over the finite field \({\mathbb {F}_n}\) all have the maximum number of \({\mathbb {F}_n}\)-rational points allowed by the Weil “explicit formulas”, and that these curves are \({\mathbb {F}_{q^2}}\)-maximal curves over infinitely many algebraic extensions \({\mathbb {F}_{q^2}}\) of \({\mathbb {F}_n}\). Serre showed that an \({\mathbb {F}_{q^2}}\)-rational curve which is \({\mathbb {F}_{q^2}}\)-covered by an \({\mathbb {F}_{q^2}}\)-maximal curve is also \({\mathbb {F}_{q^2}}\)-maximal. This has posed the problem of the existence of \({\mathbb {F}_{q^2}}\)-maximal curves other than the Deligne–Lusztig curves and their \({\mathbb {F}_{q^2}}\)-subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n 3 with n = p r > 2, p ≥ 2 prime, we give a simple, explicit construction of an \({\mathbb {F}_{q^2}}\)-maximal curve \({\mathcal {X}}\) that is not \({\mathbb {F}_{q^2}}\)-covered by any \({\mathbb {F}_{q^2}}\)-maximal Deligne–Lusztig curve. Furthermore, the \({\mathbb {F}_{q^2}}\)-automorphism group Aut\({(\mathcal {X})}\) has size n 3(n 3 + 1)(n 2 − 1)(n 2 − n + 1). Interestingly, \({\mathcal {X}}\) has a very large \({\mathbb {F}_{q^2}}\)-automorphism group with respect to its genus \({g = \frac{1}{2}\,(n^3 + 1)(n^2 - 2) + 1}\).

Unirationality of RDP Del Pezzo surfaces of degree 2

Rational points and Galois points for a plane curve over a finite field

In 2022, S. D. Cohen and the two authors introduced and studied the concept of (r,n)-freeness on finite cyclic groups G for suitable integers r, n, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of G. Combining this machinery with some character sum techniques, they explored the existence of points (x 0 ,y 0 ) on affine curves y n =f(x) defined over a finite field 𝔽 whose coordinates are generators of the multiplicative cyclic group 𝔽 * . In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension 𝔼 of a finite field 𝔽 with Q elements is a cyclic 𝔽[x]-module induced by the Frobenius automorphism α↦α Q , and any generator of this module is said to be a normal element over 𝔽. We introduce and study the concept of (f,g)-freeness on this module structure for suitable polynomials f,g∈𝔽[x]. As a main application of the machinery developed in this paper, we study the existence of 𝔽 p n -rational points in the Artin–Schreier curve 𝔄 f :y p -y=f(x) whose coordinates are normal over the prime field 𝔽 p and establish concrete results.

Number of points on certain hyperelliptic curves defined over finite fields

Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. A strong impetus in the asymptotic theory has come from a thorough exploitation of the method of infinite class field towers. On the other hand, we show by a counterexample that Perret’s conjecture on infinite class field towers is wrong, and so Perret’s method of infinite ramified class field towers breaks down. In the last two sections of the paper we discuss applications of algebraic curves over finite fields with many rational points to coding theory and to the construction of low-discrepancy sequences.

We introduce algebraic geometric techniques in secret sharing and in secure multi-party computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) defined over a finite field ${\mathbb F}_q$ , with the following properties. 1. It is ideal. The number of players n can be as large as $\#C({\mathbb F}_q)$ , where C is an algebraic curve C of genus g defined over ${\mathbb F}_q$ . 2. It is quasi-threshold: it is t-rejecting and t+1+2g-accepting, but not necessarily t+1-accepting. It is thus in particular a ramp scheme. High information rate can be achieved. 3. It has strong multiplication with respect to the t-threshold adversary structure, if $t<\frac{1}{3}n-\frac{4}{3}g$ . This is a multi-linear algebraic property on an LSSS facilitating zero-error multi-party multiplication, unconditionally secure against corruption by an active t-adversary. 4. The finite field ${\mathbb F}_q$ can be dramatically smaller than n. This is by using algebraic curves with many ${\mathbb F}_q$ -rational points. For example, for each small enough ε, there is a finite field ${\mathbb F}_q$ such that for infinitely many n there is an LSSS over ${\mathbb F}_q$ with strong multiplication satisfying $(\frac{1}{3}- \epsilon) n\leq t < \frac{1}{3}n$ . 5. Shamir’s scheme, which requires n>q and which has strong multiplication for $t<\frac{1}{3}n$ , is a special case by taking g=0. Now consider the classical (“BGW”) scenario of MPC unconditionally secure (with zero error probability) against an active t-adversary with $t<\frac{1}{3}n$ , in a synchronous n-player network with secure channels. By known results it now follows that there exist MPC protocols in this scenario, achieving the same communication complexities in terms of the number of field elements exchanged in the network compared with known Shamir-based solutions. However, in return for decreasing corruption tolerance by a small ε-fraction, q may be dramatically smaller than n. This tolerance decrease is unavoidable due to properties of MDS codes. The techniques extend to other models of MPC. Results on less specialized LSSS can be obtained from more general coding theory arguments.

Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. A strong impetus in the asymptotic theory has come from a thorough exploitation of the method of infinite class field towers. On the other hand, we show by a counterexample that Perret’s conjecture on infinite class field towers is wrong, and so Perret’s method of infinite ramified class field towers breaks down. In the last two sections of the paper we discuss applications of algebraic curves over finite fields with many rational points to coding theory and to the construction of low-discrepancy sequences.

Although the bilinear complexity of a bilinear map ϕ over a finite field may not be the minimum number of multiplications and divisions necessary for computing ϕ, the study of such maps gives some insight into the problem of computing a bilinear map over the ring of integers of a global field, such as the ring Z of integers: any bilinear computation defined over Z (that is, whose coefficients belong to Z) gives via reduction of constants modulo a prime p a bilinear computation over the finite field F p . In this chapter we introduce a relationship observed by Brockett and Dobkin [80] between the rank of bilinear maps over a finite field and the theory of linear error-correcting codes. More precisely, we associate to any bilinear computation of length r of a bilinear map over a finite field a linear code of block length r; the dimension and minimum distance of this code depend only on the bilinear map and not on the specific computation. The question about lower bounds for r can then be stated as the question about the minimum block length of a linear code of given dimension and minimum distance. This question has been extensively studied by coding theorists; we use their results to obtain linear lower bounds for different problems, such as polynomial and matrix multiplication. In particular, following Bshouty [85, 86] we show that the rank of n x n-matrix multiplication over F2 is 5/2n2 — o(n2). In the last section of this chapter we discuss an interpolation algorithm on algebraic curves due to Chudnovsky and Chudnovsky [110]. Combined with a result on algebraic curves with many rational points over finite fields, this algorithm yields a linear upper bound for R(F q n/F q )for fixed q.

In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered.