Equations for endomorphisms of abelian varieties
type: Item
instance of: scholarly article
title: Equations for endomorphisms of abelian varieties
zbMATH Open document ID: 0616.14036
DOI: 10.1007/BF01450080
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164388
review text: Let \(A\) be an abelian variety of dimension g over the field of complex numbers and \(f\) an endomorphism of \(A\). If \(A\) is embedded in a projective space \({\mathbb{P}}_ N\), then locally, that is on an open dense set \(U\subset A\), f can be described by \(N+1\) homogeneous polynomials of some degree \(n\) in \(N+1\) variables. If this can be done for a covering of A by open sets U with the \(same\quad n,\) one says that \(f\) can be described completely by forms of degree \(n.\) It is the aim of the paper to give lower bounds for such integers n. The main result is: Let F denote an ample line bundle on A and \(d\geq 3\). Then \(L=F^ d\) is very ample. Let \(A\hookrightarrow {\mathbb{P}}_ N\) be the associated complete embedding. If \(a_ 1,...,a_ g\) are the eigenvalues of f'f with respect to the analytic representation, where \(f'\) denotes the Rosati adjoint of \(f\), and \(n_ 0=[\max (a_ 1,...,a_ g)]\), then \(f\) can be described completely by forms of degree \(n_ 0+1\) and for general such \(L\) not by forms of degree \(n_ 0\). If \(A\hookrightarrow {\mathbb{P}}_ N\) is a general projectively normal embedding, the result is a little less precise: It can only be shown that f can be described completely by forms of degree \(n_ 0+2.\) The result is a common generalization of a theorem of Serre on the map \(n_ A: A\to A,\) \(n\in {\mathbb{Z}}\) [see \textit{M. Waldschmidt}, ''Nombres transcendants et groupes algébriques'', Astérisque 69-70 (1976; Zbl 0428.10017); appendix 2 by \textit{J.-P. Serre} (p. 191-202)] and a theorem on the existence of addition laws on A [the author and \textit{W. Ruppert}, Invent. Math. 79, 603-610 (1985; Zbl 0577.14035)]. In particular there are some applications to transcendental number theory.
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14E99
zbMATH DE Number: 3997991
zbMATH Keywords: abelian variety
zbMATH Keywords: Rosati adjoint
zbMATH Keywords: addition laws
zbMATH Keywords: description of endomorphism by forms
author: Herbert Lange
MaRDI profile type: Publication
cites work: Q4143433
cites work: Theta Relations and Projective Normality of Abelian Varieties
cites work: Commutative algebraic groups and intersections of quadrics
cites work: Complete systems of addition laws an abelian varieties
cites work: Q5628266
cites work: On the equations defining Abelian varieties. I-III
cites work: Some results in the theory of vector bundles
OpenAlex ID: W2074275972
description: scientific article; zbMATH DE number 3997991