p-cycles, S2-sets and Curves with Many Points

Abstract

We construct S2-sets contained in the integer interval Iq-1:= (1,q-1) with q = pn, p a prime number and n E Z+, by using the p-adic expansion of integers. Such sets come from considering p-cycles of length n. We give some criteria in particular cases which allow us to glue them to obtain good S2-sets. After that we construct algebraic curves over the finite field Fq with many rational points via minimal (Fp, Fp) -polynomials whose exponent set is an S2-set.