2020/01/28 by Dimitrov, Vesselin, Gao, Ziyang, Habegger, Philipp · 10 citations
#11G30 #11G50 #14G05 #14G25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2001.10276
Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in g and d, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for 1-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.