Let be a fixed algebraic variety defined by polynomials in variables with integer coefficients. We show that there exists a constant such that for almost all primes for all but at most points on the reduction of modulo at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.
Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants...
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.