Equations for endomorphisms of abelian varieties
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.
scientific article; zbMATH DE number 3997991