Images of finite-rank subgroups of abelian varieties under morphisms to projective space satisfy E(X) ≪ |X|^2 and |X+X| ≫ |X|^2 in affine charts, showing additive rigidity without a simplicity assumption.
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The authors derive rank-dependent uniform bounds for general patterns in the images of finite-rank subgroups of elliptic curves under maps to the projective line, extending results on Bremner's conjecture.
Direct proof via height-uniform Mordell theorem shows uniform rank bounds for elliptic curves over Q imply uniform bounds on lengths of arithmetic progressions in x-coordinates of rational points, with extensions to multiplicative groups and geometric progressions.
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Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General Case
Images of finite-rank subgroups of abelian varieties under morphisms to projective space satisfy E(X) ≪ |X|^2 and |X+X| ≫ |X|^2 in affine charts, showing additive rigidity without a simplicity assumption.
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Patterns on elliptic curves beyond Bremner's conjecture
The authors derive rank-dependent uniform bounds for general patterns in the images of finite-rank subgroups of elliptic curves under maps to the projective line, extending results on Bremner's conjecture.
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A note on Bremner's conjecture and uniformity
Direct proof via height-uniform Mordell theorem shows uniform rank bounds for elliptic curves over Q imply uniform bounds on lengths of arithmetic progressions in x-coordinates of rational points, with extensions to multiplicative groups and geometric progressions.