Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
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2 Answers
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As a supplement to Daniel's answer, I'd like to mention that for (smooth projective geometrically irreducible) curves over number fields, "Brauer-Manin is the only obstruction to the existence of rational points" is a question formulated in Skorobogatov's book (page 133) and a conjecture of mine. See my paper, in particular sections 8 and 9. The statement is true when the Tate-Shafarevich group and the Mordell-Weil group of the Jacobian are both finite, see Thm. 8.6 in the paper. Poonen in this paper (Experiment. Math. 2006) has a conjecture based on heuristic considerations that would imply the general statement.
answered Mar 17, 2015 at 11:56
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I consider only smooth curves for simplicity. In which case this is expected to be true, but certainly not known in general. In fact, it is even expected that the Brauer-Manin obstruction is already enough. For curves of genus $1$ this would follow from the finiteness of the Tate-Shafarevich group.
See Section 6.2 of the book
Skorobogatov - Torsors and rational points.
answered Mar 17, 2015 at 8:52