This is a revised and slightly expanded version. We point out that in the previous summary, "without cohomology" should really read "almost without cohomology" because of the proof of Lemma 2, that the idea to consider effective motives divisible by the Lefschetz motive was anticipated by Serre in letters to Gilles Lachaud and Marc Perret, and finally that the birational invariance of the number of points modulo q in fact follows from a 1983 Comptes Rendus note of Torsten Ekedahl (I am grateful to Antoine Chambert-Loir for making this observation and indicating this reference).
We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.
We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that there is a rational point on any degeneration of a smooth proper rationally chain connected variety over a finite field. We also obtain a generalisation of the Chevalley-Warning theorem.
One extends P. Deligne's notion of integrality over a finite field for a $\ell$-adic sheaf on a scheme of finite type over a local field with finite residue field. One shows that this integrality notion is preserved by $Rf_!$, as it is over a finite field after P. Deligne.
We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variet…
Let M be a field of finite type over {\bf Q} and X a variety defined over M. We study when the set {P \in X(K) \mid f^{\circ n} (P) = P for some n \geq 1} is finite for any finite extension fields K of M and for any dominant K-morphisms f : X \to X with deg f \geq 2.
We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p>5$. As a consequence, any klt Fano threefold over a finite field has a rational point.
Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.
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 $B$ for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on $C$ over some long rectangles is Gaussian.
Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the Störh-Voloch theory and sets new bounds to the number of $\mathbb{F}_{q^r}$-rational points on $\mathcal{X}$. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Störh-Voloch's and Ihara's.
If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
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 $f_A(1-t)$.
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.
We give a positive answer to the questions 4.4 on Hodge type and 4.2 on divisibility of the eigenvalues of Frobenius of the article by N. Fakhruddin and C. S. Rajan (NT/0402230).
Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.
We study the Fourier transforms of indicator functions of some special high-dimensional finite type domains and obtain estimates of the associated lattice point discrepancy.
We study the lattice point problem associated with a special class of high-dimensional finite type domains via estimating the Fourier transforms of corresponding indicator functions.
We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.
We find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely pure power maps z^d and Chebyshev polynomials.
In this paper we consider finite hybrid point sets that are the digital analogs to finite hybrid point sets introduced by Kritzer. Kritzer considered hybrid point sets that are a combination of lattice point sets and Hammersley point sets constructed using the ring of integers and the field of rational numbers. In this paper we consider finite hybrid point sets whose components stem from Halton-type Hammersley Point sets and lattice point sets which are constructed using the arithmetic of the ring of polynomials and the field of rational functions over a finite field. We present existence resu…
It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo $p$. In this paper, we extend this result, due to Igusa, to a family of monomial deformations of a diagonal hypersurface. We find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues.
In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $\mathbb{F}_q$-isogeny class. This work completes partial classification given by S.Rybakov.
Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.
In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results obtained improve the known bounds for a few pairs (q, g).
Let K be a number field with algebraic closure K-bar, let S be a finite set of places of K containing the archimedean places, and let f be a Chebyshev polynomial. We prove that if a in K-bar is not preperiodic, then there are only finitely many preperiodic points b in K-bar which are S-integral with respect to a.
We produce a collection of families of curves, whose point count statistics over F_p becomes Gaussian for p fixed. In particular, the average number of F_p points on curves in these families tends to infinity.
Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves constructed are Kummer covers or fibre products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves.
We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many points a curve over GF(2) of genus 17 can have, by finding one with 18 points. The search is aided by computer; in some cases it is exhaustive for this type of curve of genus up to 50.
Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we characterise for which finite field extensions $\mathbb{F}_{q^k}$, $k\geq 1$ (if any) the corresponding groups of $\mathbb{F}_{q^k}$-rational points are isomorphic, i.e. $E(\mathbb{F}_{q^k}) \cong E'(\mathbb{F}_{q^k})$.
We show that there is no simple congruence formula for the number of points of the mod p reduction of a regular model of a smooth proper variety defined over a local field with $\ell$-adic cohomology supported in codimension $\ge κ$.
We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the limits taken differ slightly and the resulting distributions are interestingly different.
The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known bounds. More precisely, we improve the lower bounds on the maximal number of points $N_q(g)$ for many values of the genus $g$ and of the cardinality $q$ of the finite field $\mathbb F_q$, by looking at coverings of all genus three smooth projective curves over $\mathbb F_q$, for $q$ is an odd prime less than 19.
We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in $\mathbb{R}^d$ ($d\geq 2$) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We prove that optimal domains which contain the most positive (or least nonnegative) lattice points are asymptotically balanced.
Let $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the dynamical Mordell-Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{N}~|~f^n(a) \in V \}$ satisfies a strong gap principle.
The main purpose of this paper is to prove that the point-line incidence bound due to Vinh (2011) over arbitrary finite fields can be improved in certain ranges by using tools from the VC-dimension theory. As consequences, a number of applications will be discussed in detail.
Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that if a point P in E(F) is nontorsion, then there are only finitely many torsion points x in E(F) which are S-integral with respect to P. We also prove an analogue of this for the multiplicative group, and formulate conjectural generalizations for abelian varieties and dynamical systems.
For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least $6q^{k-3}$ rational points over $\mathbb{F}_{q}$ if $1\leq m \leq q-3$ and $q$ is odd. A key ingredient in our proof is Segre's classical theorem on ovals in finite projective planes.
We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average, less than two searches suffice to obtain a rational point. We also analyze the probability distribution of outputs, using the notion of Shannon entropy, and prove that the algorithm is somewhat close to any "ideal" equidistributed algorithm.
