On the order of points on curves over finite fields.
On certain plane curves with many integral points.
Lang's Conjectures, Fibered Powers, and Uniformity.
Diophantine Approximation in Characteristic p.
Chebyshev's method for number fields
We give an elementary proof of an explicit estimate for the number of primes splitting completely in an extension of the rationals. The proof uses binomial coefficents and extends Chebyshev's classical approach.
An analogue of the Weierstrass ζ-function in characteristic p
On the p-adic Waring's problem
On some subgroups of the multiplicative group of finite rings
Let be a subset of , the field of elements and a polynomial of degree with no roots in . Consider the group generated by the image of in the group of units of the ring . 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...
Companion forms and Kodaira-Spencer theory.
Reduction of the Manin map modulo p.
Lang’s conjecture in characteristic : an explicit bound
A formula for the Cartier operator on plane algebraic curves.
Integral points of abelian varieties over function fields of characteristic zero.
A local-global criterion for dynamics on ℙ¹
On the number of places of convergence for Newton’s method over number fields
Let be a polynomial of degree at least 2 with coefficients in a number field , let be a sufficiently general element of , and let be a root of . We give precise conditions under which Newton iteration, started at the point , converges -adically to the root for infinitely many places of . As a corollary we show that if is irreducible over of degree at least 3, then Newton iteration converges -adically to any given root of for infinitely many places . We also conjecture that...
Visible Points on Curves over Finite Fields
For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula 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.
Diophantine approximation and deformation
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