Abelian varieties with several principal polarizations
type: Item
instance of: scholarly article
title: Abelian varieties with several principal polarizations
zbMATH Open document ID: 0657.14023
DOI: 10.1215/S0012-7094-87-05531-1
author: Herbert Lange
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: 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.
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 4073164
zbMATH Keywords: abelian variety
zbMATH Keywords: principal polarizations
zbMATH Keywords: symmetric automorphisms
zbMATH Keywords: real multiplication
zbMATH Keywords: class number
MaRDI profile type: Publication
cites work: Classnumbers and unit signatures
cites work: On Totally Real Cubic Fields
cites work: Existence of curves of genus two on a product of two elliptic curves
cites work: A class number associated with the product of an elliptic curve with itself
cites work: Q2914641
cites work: Über Den Bizyklischen Biquadratischen Zahlkörper
cites work: Joacobian surfaces in P4 .
cites work: Q5628266
cites work: Polarisations on an abelian variety
cites work: Q3943047
full work available at URL: s0012-7094-87-05531-1
OpenAlex ID: W2026616912
description: scientific article; zbMATH DE number 4073164