Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].     

Improved upper bounds for the number of rational points on algebraic curves over finite fields

Currently, the best known bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g, defined over a finite field %plane1D;53D;_q, generally come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from the optimization of the explicit formulae if the genus is large. We give methods for improving these bounds in both cases. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 8, 32, 2¹³, 27, 243, 125, and when q = 2^s, s > 1. For large genera, isolated improvements are obtained for q = 3, 8, 9.     

Bounds on the number of automorphisms of curves over algebraically closed fields

In this note we compare the maximum number of automorphisms of smooth projective curves of genus g ≥ 2 over algebraically closed fields of characteristic 0 or a prime p > 0, respectively. For given g ≥ 2, we show that for almost all primes p > 0, this number is the same.     

The distribution of the number of points modulo an integer on elliptic curves over finite fields

Let $\mathbb{F}_{q}$ be a finite field, and let b and N be integers. We prove explicit estimates for the probability that the number of rational points on a randomly chosen elliptic curve E over $\mathbb{F}_{q}$ equals b modulo N. The underlying tool is an equidistribution result on the action of Frobenius on the N-torsion subgroup of E. Our results subsume and extend previous work by Achter and Gekeler.     

Number of points on certain hyperelliptic curves defined over finite fields

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums _l(v), ψ_l(v), and ψ_2l(v), and the Jacobsthal–Whiteman sums and , over finite fields F_q such that . These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n1, the exact number of F_qⁿ-rational points on the projective hyperelliptic curves aY²Z^e−2=bX^e+cZ^e (abc≠0) (for e=l,2l), and aY²Z^l−1=X(bX^l+cZ^l) (abc≠0), defined over such finite fields F_q. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over F_q, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F_q.     

Finite number of fibre products of Kummer covers and curves with many points over finite fields

We study fibre products of a finite number of Kummer covers of the projective line over finite fields. We determine the number of rational points of the fibre product over a rational point of the projective line, which improves the results of Özbudak and Temür (Appl Algebra Eng Commun Comput 18:433–443, 2007) substantially. We also construct explicit examples of fibre products of Kummer covers with many rational points, including a record and two new entries for the current table (http://www.manypoints.org, 2011).     

Rational points and zeta functions of some curves over finite fields

In this paper we study the number of rational points on some curves over finite fields. Moreover, zeta functions of the associated function fields are evaluated explicitly.     

The genus of curves over finite fields with many rational points

We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field and whose number of -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)² or 2g=(q−1)q. Supported by a grant from the International Atomic Energy and UNESCOCorrespondence to: F. Torres This article was processed by the author using theLatex style file from Springer-Verlag.     

Concentration of points on curves in finite fields

We obtain analogues of several recent bounds on the number of solutions of polynomial congruences modulo a prime with variables in short intervals in the case of polynomial equations in high degree extensions of finite fields. In these settings low-dimensional affine spaces play the role of short intervals and thus several new ideas are required.     

The distribution of points on curves over finite fields in some small rectangles

Let \(p\) be a prime. We study the distribution of points on a class of curves \(C\) over \(\mathbb{F }_p\) inside very small rectangles \(\mathcal{B }\) for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on \(C\) over long rectangles is Gaussian.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields

In this paper, we describe an algorithm for computing the order of the Jacobian varieties of Picard curves over finite fields. This is an extension of the algorithm of Matsuo, Chao and Tsujii (MCT) [K. Matsuo, J. Chao, S. Tsujii, An improved baby step algorithm for point counting of hyperelliptic curves over finite fields, in: LNCS vol. 2369, Springer-Verlag, 2005, pp. 461–474] for hyperelliptic curves. We study the characteristic polynomials and the Jacobian varieties of algebraic curves of genus three over finite fields. Based on this, we investigate the explicit computable bounds for coefficients of the characteristic polynomial and improve a part of the baby-step giant-step of the counting points algorithm. Usefulness of the proposed method is illustrated and verified by the simple examples.