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Reduction of abelian varieties and curves

Moshe Jarden|Aharon Razon
대한수학회 (The Korean Mathematical Society)|2024.05|대한수학회지 (Journal of the KMS)|제61권 제3호|pp. 515-545|DOI 10.4134/JKMS.j230367

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키워드

Reductionmoduli spaceabelian varietyisotriviality
주제분류수학

초록

Consider a Noetherian domain $R_0$ with quotient field $K_0$. Let $K$ be a finitely generated regular transcendental field extension of $K_0$. We construct a Noetherian domain $R$ with $\Quot(R)=K$ that contains $R_0$ and embed $\Spec(R_0)$ into $\Spec(R)$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\frp$ for ``almost all'' $\frp\in\Spec(R_0)$.

Abstract

Consider a Noetherian domain $R_0$ with quotient field $K_0$. Let $K$ be a finitely generated regular transcendental field extension of $K_0$. We construct a Noetherian domain $R$ with $\Quot(R)=K$ that contains $R_0$ and embed $\Spec(R_0)$ into $\Spec(R)$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\frp$ for ``almost all'' $\frp\in\Spec(R_0)$.

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