Analysis highlights that the conjugacy class of Selmer sections is uniquely determined, indicating significant properties regarding cuspidal reductions.
We consider reductions of Selmer sections of the étale homotopy sequence of a hyperbolic curve over a number field. We show that the conjugacy class of a noncuspidal Selmer section is uniquely determined by its reduction on a set of density one. Moreover, we show that a noncuspidal Selmer section reduces to cusps only on a set of density zero, at least after a finite field extension.
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Wojciech Porowski (2025) studied this question.
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