Conditional Results on the Birational Section Conjecture over Small Number Fields

In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speaking, for any birational Galois section of the projective line, the local points associated to the birational Galois section avoid distinct three rational points, and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes. Moreover, as another consequence, we obtain some examples of projective smooth curves for which any prosolvable birational Galois section is geometric.