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.