A decision procedure for certain abelian varieties over function fields
type: Item
instance of: scholarly article
title: A decision procedure for certain abelian varieties over function fields
zbMATH Open document ID: 0807.14033
published in: Journal of Algebra
publication date: 1995-03-05T00:00:00Z
review text: Let \(\mathbb{Q}_ c\) be the algebraic closure of the rational numbers. The authors prove the existence of a decision procedure for the additive group of isogenies between two abelian varieties over \(\mathbb{Q}_ c (t)\). Using Faltings' isogeny theorem, a countable listing of possible isogenies for a sequence of lower bounds, and a calculation of Galois actions on the division points modulo \(m\) for increasing \(m\) for an upper bound, one gets a decision procedure for abelian varieties which become trivial over some finite extension fields. A formula of Kodaira and Shioda for the rank of general elliptic surfaces is proven to be valid for an abelian varieties over \(\mathbb{Q}_ c (t)\) having no continuous family of sections, namely the rank of numerical equivalence classes of 2-dimensional divisors modulo classes from fibres and the zero section. The paper also gives a computable upper bound on the rank based on étale cohomology which is equal to the rank if the Tate conjecture holds.
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 554215
zbMATH Keywords: isogenies between abelian varieties
zbMATH Keywords: decision procedure
author: Ki Hang Kim
author: Fred Roush
MaRDI profile type: Publication
full work available at URL: jabr.1994.1026
OpenAlex ID: W2010968629
DOI: 10.1006/JABR.1994.1026
description: scientific article; zbMATH DE number 554215