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The Rank of Abelian Varieties over Infinite Galois Extensions

โœ Scribed by Michael Rosen; Siman Wong

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
151 KB
Volume
92
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

asked if every Abelian variety A defined over a number field k with dim A > 0 has infinite rank over the maximal Abelian extension k ab of k. We verify this for the Jacobians of cyclic covers of P 1 , with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d-dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).

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