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Period and index of a curve $X/K$ over a $p$-adic local field $K$ such thatthe fundamental group $\pi_1(X/K)$ admits a splitting are shown to be powers of$p$. As a consequence, examples of curves over number fields are constructedwhere having sections is obstructed locally at a $p$-adic place. Hence thesection conjecture holds for these curves as there are neither sections norrational points.

On the period-index problem in light of the section conjecture