In a principally polarized abelian variety A isogenous to a product of two factors X and Y, the types of the induced polarizations on X and Y determine each other. This fact makes it possible to explicitly construct a period matrix for A, given only the period matrices of the two complementary abelian subvarieties, see [<CitationRef CitationID="CR1">1</CitationRef>]. Unfortunately, the method cannot be adapted to the case when having more factors in a decomposition of A or when A is not principally polarized. Here we present a result, which leads into an algorithm, to recover a Riemann matrix for a polarized abelian variety A decomposed via an isogeny <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1644_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="170" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi :A_1\times \cdots \times A_r \rightarrow A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>A</mi> <mi>r</mi> </msub> <mo stretchy="false">→</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> if the period matrices for each subvariety <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1644_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$A_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> and the rational representation of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1644_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> are known. We show with an example how this method works.

Finding a period matrix for a decomposed Abelian variety

期刊论文
发表日期：

2025年6月10日

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Rubí E. Rodríguez,
Anita M. Rojas

摘要

In a principally polarized abelian variety A isogenous to a product of two factors X and Y, the types of the induced polarizations on X and Y determine each other. This fact makes it possible to explicitly construct a period matrix for A, given only the period matrices of the two complementary abelian subvarieties, see [1]. Unfortunately, the method cannot be adapted to the case when having more factors in a decomposition of A or when A is not principally polarized. Here we present a result, which leads into an algorithm, to recover a Riemann matrix for a polarized abelian variety A decomposed via an isogeny $\varphi :A_1\times \cdots \times A_r \rightarrow A$ $φ : A_{1} \times \dots \times A_{r} \to A$ if the period matrices for each subvariety $A_j$ $A_{j}$ and the rational representation of $\varphi $ $φ$ are known. We show with an example how this method works.