Exponential sums over definable subsets of finite fields

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates of varieties of finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application, there is no formula in the language of rings that defines for infinitely many primes an “interval” in Z/p Z that is neither bounded nor with bounded complement.     

On some definable sets over fields with analytic structure

We discover geometric properties of certain definable sets over non-Archimedean valued fields with analytic structures. Results include a parameterized smooth stratification theorem and the existence of a bound on the piece number of fibers for these sets. In addition, we develop a dimension theory for these sets and also for the formulas which define them.     

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.     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ mathbb{F}mathfrak{M} $ mathbb{F}mathfrak{M} over a model $ mathfrak{M} $ mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

Congruences for rational points on varieties over finite fields

We prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial Chow group of 0-cycles. We also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the Chow group of 0-cycles of the generic fibre is trivial. In particular this leads to a vast generalization of the classical Chevalley-Warning theorem. The above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of H. Esnault on the number of rational points of smooth proper varieties over finite fields with trivial Chow group of 0-cycles.     

Factorization formulae on counting zeros of diagonal equations over finite fields

Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

Classes of points of cubic hypersurfaces defined over finite fields

The points of a cubic hypersurface are divided into equivalence classes of Ya. I. Manin [1]. It is established that the quasigroup of these equivalence classes over a finite field is isomorphic to either 1 or Z₂ or Z₃ under some weak conditions.     

The groups of points on abelian varieties over finite fields

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f _A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f _A (1 − t).     

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane over a finite field F_q, where the formally defined squared Euclidean distance of every pair of points is a square in F_q. It turns out that integral point sets over F_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order.In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F_q for q≤47. Furthermore, we give two series of maximal integral point sets and prove their maximality.     

An asymptotic formula for counting subset sums over subgroups of finite fields

Let ${\mathbb{F}}_q$ be the finite field of q elements. Let $H?{\mathbb{F}}_q^{?}$ be a multiplicative subgroup. For a positive integer k and element $b\in {\mathbb{F}}_q$, we give a sharp estimate for the number of k-element subsets of H which sum to b.     

Large sets of t-designs over finite fields exist for all t

Designs, Codes and Cryptography - A t- $$(n,k,\lambda )_q$$ design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field $$\mathbb {F}_q$$...     

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.     

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.     

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.