Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture


Authors: Richard Pink and Damian Roessler
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 771-798
DOI: https://doi.org/10.1090/S1056-3911-04-00368-6
Published electronically: February 11, 2004
MathSciNet review: 2073195
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Abstract | References | Additional Information

Abstract: Let $A$ be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism $\psi: A\to A$. In this paper, we give an essentially complete classification of all $\psi$-invariant subvarieties of $A$. For example, under some mild assumptions on $(A,\psi)$ we prove that every $\psi$-invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of $K$. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.


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Additional Information

Richard Pink
Affiliation: Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland
Email: pink@math.ethz.ch

Damian Roessler
Affiliation: Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland
Email: roessler@math.ethz.ch

DOI: https://doi.org/10.1090/S1056-3911-04-00368-6
Received by editor(s): July 17, 2002
Received by editor(s) in revised form: September 13, 2002
Published electronically: February 11, 2004

American Mathematical Society