2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155838Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $π_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points.17 pagesAlgebraic GeometryNumber Theory14H30; 14G05text