Details
| Original language | English |
|---|---|
| Pages (from-to) | 1509–1545 |
| Number of pages | 37 |
| Journal | Mathematische Annalen |
| Volume | 377 |
| Issue number | 3-4 |
| Early online date | 17 Jun 2020 |
| Publication status | Published - Aug 2020 |
| Externally published | Yes |
Abstract
Fix an elliptic curve E 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 E0g, also defined over the algebraic numbers, under all isogenies between E0g 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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Mathematische Annalen, Vol. 377, No. 3-4, 08.2020, p. 1509–1545.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Unlikely intersections with isogeny orbits in a product of elliptic schemes
AU - Dill, Gabriel A.
N1 - Publisher Copyright: © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/8
Y1 - 2020/8
N2 - Fix an elliptic curve E 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 E0g, also defined over the algebraic numbers, under all isogenies between E0g 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.
AB - Fix an elliptic curve E 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 E0g, also defined over the algebraic numbers, under all isogenies between E0g 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.
UR - http://www.scopus.com/inward/record.url?scp=85086595685&partnerID=8YFLogxK
U2 - 10.1007/s00208-020-02024-2
DO - 10.1007/s00208-020-02024-2
M3 - Article
VL - 377
SP - 1509
EP - 1545
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 3-4
ER -