Generalized Vojta–Rémond inequality

Dill, Gabriel Andreas

doi:10.1142/s1793042120500062

Cited by 2 publications

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References 12 publications

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“…, X m be a family of irreducible positive-dimensional projective varieties, defined over Q. In the present work, we use a generalization [7] of Rémond's results of [52] to the case of an algebraic point x = (x 1 , . .…”

Section: Appendix a Generalized Vojta-rémond Inequalitymentioning

confidence: 99%

“…In Section 3, we show that it suffices to prove Conjecture 1.2 for V of a certain non-degenerate type without placing any restrictions on A. In Section 4, we apply a generalized Vojta-Rémond inequality (see Appendix A and [7]) to deduce a height bound of the necessary form for a sufficiently large subset of A Γ ∩ V if V is not degenerate and A and A 0 are of the form described in Theorem 1.3. In Section 5, we apply the Pila-Zannier strategy and use the height bound we obtained in Section 4.…”

Section: Introductionmentioning

confidence: 99%

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Unlikely intersections with isogeny orbits in a product of elliptic schemes

Dill

2019

Preprint

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Fix an elliptic curve E0 without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of E g 0 , also defined over the algebraic numbers, under all isogenies between E g 0 and some fiber of the g-th fibered power A of the elliptic scheme, where g is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside A that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-Rémond inequality with the Pila-Zannier strategy. Contents 1. Introduction 1 2. Preliminaries and notation 5 3. Reduction to the non-degenerate case 6 4. Height bounds 9 5. Application of the Pila-Zannier strategy 26 6. Proof of Theorem 1.3 30 Acknowledgements 31 Appendix A. Generalized Vojta-Rémond inequality 32 References 33

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Section: Appendix a Generalized Vojta-rémond Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%