Abelian varieties with several principal polarizations
Let \(A\) be an abelian variety over the complex numbers, the author studies the set \(\Pi(A)\) of isomorphism classes of principal polarizations of \(A\), which has finite cardinality \(\pi(A)\) by a result of Narasimhan and Nori. Assuming there exists one principal polarization \(L_ 0\), the author proves (theorem 1.5): \(L_ 0\) induces a bijection between \(\Pi(A)\) and the set of classes of totally positive \(L_ 0\)-symmetric automorphisms of \(A\) modulo \(\text{Aut}(A)\). (An endomorphism of \(A\) is totally positive if the zeroes of its minimal polynomial are positive.) In case \(A\) has real multiplication by an order in the totally real number field \(K\), the author further gives criteria for \(\pi (A)>0\), and the estimate (theorem 3.1) \(\pi (A)\leq h^+/h\), which is sharp if \(\text{End}(A)\) is the maximal order of \(K\) (\(h,h^+\) = class number, narrow class number of \(K\), respectively). The paper concludes with some very interesting examples, based on the above mentioned statements and numerical data of several other authors.
scientific article; zbMATH DE number 4073164
scientific article