We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.
A planar surface is a surface in three-space in which every tangent line has triple or higher contact with the surface at the point of tangency. We study properties of planar surfaces in positive characteristics, use that to bound the number of points of a planar surface over a finite field and give an application to Waring's problem for polynomials.
Using the relation between the problem of counting irreducible polynomials over finite fields with some prescribed coefficients to the problem of counting rational points on curves over finite fields whose function fields are subfields of cyclotomic function fields, we count the number of generators of finite fields with powers of trace zero up to some point, answering a question of Z. Reichstein.
For a morphism of a variety X over a number field K, we consider local conditions and a "Brauer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P and V. This evidence stems from a probabilistic argument as well as unconditional results in the case of étale maps.
We give an explicit description of the F_{qi}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0, for i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{qi}. We also describe the F_{q2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0.
We study a new obstruction to the existence of integral and rational points for algebraic varieties over function fields, the differential descent obstruction. We prove that that is the only obstruction to the existence of integral points in affine varieties in characteristic zero and also, in most cases, for rational points on curves in arbitrary characteristic.
This paper supersedes an earlier version by the second author which had several erroneous claims and incomplete results We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. For the latter, we prove some partial results that indicate that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.
We give examples of sequences of smooth non-isotrivial curves for every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence cannot be uniformly bounded.
We prove that the Brauer-Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic p. More precisely, we show that the only obstructions come from etale covers of exponent p or, alternatively, from flat covers coming from torsors under connected group schemes of exponent p.
We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of integral points on integral models of twists of modular curves over function fields.
We prove estimates on character sums on the subset of points of an elliptic curve over F_{Q} with x-coordinate of the form α + t where t varies in F_{q} and α is fixed such that F_{Q} = F_{q}(α). We deduce that, for a suitable choice of α this subset has a point of maximal order in E(F_{Q}). This provides a deterministic algorithm for finding a point of maximal order which for a very wide class of finite fields is faster than other available algorithms.
We present a technique based on bounds of character sums to prove the indifferentiability of hash function constructions based on essentially any deterministic encoding to elliptic curves or curves of higher genus, such as the algorithms by Shallue, van de Woestijne and Ulas, or the Icart-like encodings recently presented by Kammerer, Lercier and Renault. In particular, we get the first constructions of well-behaved hash functions to Jacobians of hyperelliptic curves. Our technique also provides more precise estimates on the statistical behavior of those deterministic encodings and the hash function constructions based on them. Additionally, we can derive pseudorandomness results for partial bit patterns of such encodings.
Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x be a sufficiently general element of K, and let z be a root of f. We give precise conditions under which Newton iteration, started at the point x, converges v-adically to the root z for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.
We prove a non-existence result for special divisors of large dimension on curves over finite fields with many points. We also give a family of examples where such divisors exist under less stringent hypotheses.
We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of S integral points on integral models of twists of modular curves over Q, for any finite set of primes S. We deduce this from an existence theorem for elliptic curves over Q satisfying certain local conditions.
We discuss the question of whether the Brauer-Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and relate it to an old conjecture of Skolem. We show that for elliptic curves minus one point the question has a negative answer.
We study the hash function from a finite field into an elliptic curve over that field which has recently been introduced by T. Icart. In particular we slightly adjust and prove the asymptotic formula conjectured by T. Icart for the image size of this function.
We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that the values on points of order small with respect to their degree of rational functions on an elliptic curve have high order. We discuss several special cases, including an old construction of Wiedemann, giving the first non-trivial estimate for the order of the elements in this construction.
Let K be a number field or a one-dimensional function field, we consider a rational map of degree at least two defined over K, and a point P in P^{1}(K) with infinite orbit under the action of the map, and Z a finite set of points. We prove a local-global criterion for the intersection of the orbit of P and the finite set Z. This is a special case of a dynamical Brauer-Manin criterion suggested by Hsia and Silverman.
Akiyama and Goto have proposed a cryptosystem based on rational points on curves over function fields (stated in the equivalent form of sections of fibrations on surfaces). It is easy to construct a curve passing through a few given points, but finding the points, given only the curve, is hard. We show how to break their original cryptosystem by using algebraic points instead of rational points and discuss possibilities for changing their original system to create a secure one.
We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields.
We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that for points on a plane curve, one of the coordinates has to have high order. We also discuss a conjecture of Poonen for subvarieties of semiabelian varieties for which our result is a weak special case. Finally, we look at some special cases where we obtain sharper bounds.
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.
We view an algebraic curve over Q as providing a one-parameter family of number fields and obtain bounds for the average value of some standard prime ideal counting functions over these families which are better than averaging the standard estimates for these functions.
For a prime p and an absolutely irreducible modulo p polynomial f(U,V) in Z[U,V] we obtain an asymptotic formulas for the number of solutions to the congruence f(x,y) = a mod p in positive integers x < X, y < Y, with the additional condition gcd(x,y)=1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.
We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
We discuss some applications of the theory of algebraic curves to the study of S-boxes in symmetric cryptography.
We study error-correcting codes constructed from projective surfaces over finite fields using the generalized Goppa construction. We obtain bounds for the minimal distance of these codes by understanding how the zero sets of functions on a surface decompose into irreducible components. We also present a decoding algorithm for these codes based on the Luby-Mitzenmacher algorithm for LDPC codes.
For an elliptic curve over a function field and a subgroup of rank at least six, we prove that the reduction of the subgroup modulo a place v covers the group of points of the curve modulo v for a positive proportion of v's.
For infinitely many primes p, the minimal distance of the binary quadratic residue code of length p is O(p/log log p).
We present an algorithm to compute r-th roots in a finite field with q^{m} elements with complexity O((log m + rlog q)m^{2}(log q)^{2}) for certain choices of m and q.
In this note we give a lower bound for the minimal distance of the double circulant binary quadratic residue codes.
We describe an algorithm that improves on the standard algorithm for computing the minimal distance of cyclic codes.
We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational points on a family of genus 2 curves over a finite field.
We construct (k,n)-arcs in PG(2,q) with k approximately q^{2}/d and n approximately q/d for each divisor d of q-1.
The main result of this paper is that, in a precise sense, a positive proportion of all hypersurfaces in P^{n} of degree d defined over Q are everywhere locally solvable, provided that n,d > 1 and (n,d) is not (2,2). This result is motivated by a conjecture discussed in detail in the paper about the proportion of hypersurfaces as above that are globally solvable, i.e., have a rational point.
Let S be a subset of F_{q}, the field of q elements and h in F_{q}[x] a polynomial of degree d>1 with no roots in S. Consider the group generated by the image of {x-s | s in S} in the group of units of the ring F_{q}[x]/(h). In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.
Contains the results of the short note "Improvements to AKS". pdf file .
We construct irreducible plane curves over finite fields with p elements, p prime, with degree near p/2 which have d(d+p-1)/2 rational points. We also prove an irreducibility criterion for plane curves.
We give a formula as an exponential sum for a homogeneous weight on Galois rings (or equivalently, rings of Witt vectors) and use this formula to estimate the weight of codes obtained from algebraic geometric codes over rings.
We give bounds for the minimal distance of duals of binary BCH codes. This is done by bounding the number of points on curves of the type y^{2}-y=f(x) over finite fields of characteristic two.
We investigate some plane curves with many points over Q, finite fields and cyclotomic fields.
In this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the p-curvatures.
We prove that a smooth surface in P^{3} of degree d, defined over a finite field with q elements, q prime, has at most d(d+q-1)(d+2q-2)/6 + d(11d-24)(q+1) rational points.
The math behind the puzzle Blet.
We give a new construction of rings of fractions (or localizations) and deduce their basic properties, the hard way.
We give an upper bound for the least prime number which does not split completely in a Galois extension of Q in terms of the degree and discriminant of the extension.
We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree.
We show that, if K/Q is a galois extension, the number of primes splitting in K is at least cx^{1/d}/log x by considering binomial coefficients.
We study whether the set of rational points of a curve over a finite field generates the set of rational points of its Jacobian. We show that this happens if the field is large enough compared to the genus. We also show that when this doesn't happen we obtain curves with many points. We give numerical examples of the latter situation which yield curves with the biggest known number of rational points for their genera.
Let f(x,y) be a polynomial of degree d in two variables whose coefficents are integers in an unramified extension of Q_{p}. Assume that the reduction of f modulo p is irreducible of degree d and not a binomial. Assume also that p > d^{2} +2. Then the number of solutions of the inequality |f(z_{1},z_{2})| < p^{-1}, with z_{1},z_{2} roots of unity in the algebraic closure of Q_{p} or zero, is at most pd^{2}.
This is a write-up of lectures presented at the first Arizona Winter School in Arithmetic Geometry on the abc conjecture.
We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing "lifts" of these curves and then estimating the size of certain exponential sums.
We study Waring's problem on unramified extensions of Z_{p}. In particular we prove that every p-adic integer is a sum of 9 pd-th powers if p is sufficiently large compared to d.
We study which subgroups of the torsion subgroup of commutative algebraic groups over finite fields can be defined by difference equations.
We prove that {(n^{p}-n)/p}_{p} in the product of all F_{p} is independent of 1 over the integers assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.
Elliptic Wieferich primes generalize the notion of Wieferich primes (primes p with p^{2} dividing n^{p}-n) to elliptic curves. We generalize a result of Granville to elliptic Wieferich primes and also study them for function fields.
We give a bound for number of points in the intersection of ax+by=1 with a finitely generated group in (K^{*})^{2}, K a field of characteristic p in terms of p and the rank of the group.
We give a bound for number of points in the intersection of a curve with a finitely generated group in the Jacobian of a curve in positive characteristics, for non-isotrivial curves.
For an abelian variety A over a function field K of characteristic zero, Manin defined a remarkable additive map (K) \ra V, where V is a vector space over K. We define an analogue of this map in the case of function fields of characteristic p. We then prove that the reduction modulo p of the Manin map in characteristic zero is the derivative of the Manin map in characteristic p and that the kernel of the Manin map in characteristic p is the group of points divisible by p.
In this paper we construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we develop some tools; notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. Another application of our construction is to obtain low-correlation sequences suitable for use in code-division multiple access (CDMA). Some pari code to compute canonical lifts and a few other things are available.
This paper shows how the construction of the previous paper works in the special case of Z/4Z and does a numerical example in detail.
We study differential operators as linear operators in power series fields, prove some of their properties (they are continuous but not differentiable) and compute their Mahler-Wagner expansion.
Abstract: We study diophantine approximations to algebraic functions in characteristic p. We precise a theorem of Osgood and give two classes of examples showing that this result is nearly sharp. One of these classes exhibits a new phenomenon.
New version.
The purpose of this note is to relate the discrete logarithm problem (DLP) on elliptic curves to descents. Let G be a group. The DLP for G is to find an procedure so that, given P,Q \in G one finds an integer m with Q=mP or shows that m does not exist. We use descents to relate the DLP on elliptic curves to the DLP on multiplicative groups in the prime to p part and additive groups for the p-part. We also discuss the relation with other approaches, in particular, the Smart-Satoh-Araki and Semaev approaches to the discrete logarithm problem on anomalous elliptic curves.
This note is now incorporated in the above paper.
We construct a sequence of polynomials P_{d} in two variables with integer coefficients that define plane curves with many integral points. Some pari code to compute these polynomials and a few other things are available.
Let C be the curve y^{2}=x^{6}+1 of genus 2 over a field of characteristic zero. Consider C embedded in its Jacobian J by sending one of the points at infinity on C to the origin of J. In this brief note we show that the points of C whose image on J are torsion are precisely the two points at infinity, the two points with x=0 and the six points with y=0.
We prove the finiteness of integral points on affine open subsets of "sufficiently general" abelian varieties over function fields of positive characteristic. We also obtain results on an abelian analogue of Leopoldt's conjecture in the same context.
We define a metric on the points of a variety defined over a non-archimedian local field and prove various properties of it.
We establish an analogue of the analytic parametrization of abelian varieties in characteristic p, which in some cases serves as an analogue and generalizes the Tate parametrization of elliptic curves over local fields with multiplicative reduction and give some applications. If K is a separably closed field of characteristic p > 0 and E/K is an ordinary elliptic curve, then \widehat {E(K)} is isomorphic to \widehat {K^*}/\Lambda, where, for an abelian group , hat A is the inverse limit of A/p^nA and \Lambda is a {\bf Z}_p-submodule of \widehat {K^*} of rank at most 1.
We prove that if a_1,...,a_n are in C_p, the completion of the algebraic closure of Q_p, there exists a constant c > 0 such that for any z_1,...,z_n roots of unity in C_p either sum z_ia_i = 0 or |sum z_ia_i| > c. The proof splits into two steps. First we show the result is true if the roots of unity are restricted to have order prime to p and the a_i are in an unramified extension of Q_p, and then we reduce the general case to that case. We will be able to say a lot more in the situation of the first step and develop an analogy with a similar problem in power series fields.
This is very short survey of Diophantine geometry in characteristic p almost without proofs.
We prove the following result: Let A be a semiabelian variety over \Cp and X a closed subvariety of A. Assume that the Frobenius endomorphism of the reduction lifts to an endomorphism of A. Then there exists c>0 such that, for every torsion point P of A, either P \in X or d(P,X) \ge c.
If K is a global field of positive characteristic and v is a place of K where an elliptic curve E has split multiplicative reduction, then the Tate parameter q of E is transcendental over K and so is any element of the completion of K at v which maps to a point of infinite order in E(K) under the Tate parametrization.
We prove that the fibered power conjecture of Caporaso, Harris and Mazur together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted by Caporaso et al. A few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many points; and that curves over k(t) can have arbitrarily many Frobenius orbits of non-constant points where k is the algebraic closure of a finite field.
Cassels has introduced an analogue for the Weierstrass zeta function (integral of the p-function) in characteristic p. We study this function. We prove an addition formula and differential equation for it. We relate it to the Mazur-Tate sigma function. Finally we use it to describe the universal vectorial extension of an elliptic curve, as done by Lang and Katz in characteristic zero.
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