Proves finiteness of reducible fibers over degree-d algebraic points for curve coverings over number fields, with a consequence for high-degree indecomposable rational functions to P1.
Points of low degree on curves over function fields
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abstract
We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for $d=1$, Harris--Silverman for $d=2$, Abramovich--Harris for $d=3,4$ and Kadets--Vogt for $d=4,5$. Our approach uses a specialization argument to reduce the problem over function fields to the number field case.
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2026 1verdicts
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Hilbert irreducibility for algebraic points
Proves finiteness of reducible fibers over degree-d algebraic points for curve coverings over number fields, with a consequence for high-degree indecomposable rational functions to P1.