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Unlikely Intersections on families of abelian varieties (Part II) Capuano, Laura
Description
Let $A$ be a non-isotrivial family of abelian varieties over a smooth irreducible curve $S$. Suppose the generic fiber of $A$ is simple and call $R$ its endomorphism ring. We consider an irreducible curve $C$ in the $n$-fold fibered power of $A$ and suppose that everything is defined over a number field $k$. Then $C$ defines $n$ points $P_1, ... P_n$ points on $A(k(C))$. Then, there are at most finitely many points $c$ on the curve such that the specialized $P_1(c), ... P_n(c)$ are dependent over $R$, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.
Item Metadata
Title |
Unlikely Intersections on families of abelian varieties (Part II)
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-11-14T10:31
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Description |
Let $A$ be a non-isotrivial family of abelian varieties over a smooth irreducible curve $S$. Suppose the generic fiber of $A$ is simple and call $R$ its endomorphism ring. We consider an irreducible curve $C$ in the $n$-fold fibered power of $A$ and suppose that everything is defined over a number field $k$. Then $C$ defines $n$ points $P_1, ... P_n$ points on $A(k(C))$. Then, there are at most finitely many points $c$ on the curve such that the specialized $P_1(c), ... P_n(c)$ are dependent over $R$, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.
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Extent |
67 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Oxford
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Series | |
Date Available |
2018-05-14
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0366266
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International