• 대한수학회지
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• 제46권2호
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• pp.237-248
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• 2009

We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered $hyperk{\ddot{a}}hler$ manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.295-300
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• 2016

For the complex torus, there is the Riemann condition on the polarization for it to be an abelian variety. We extend this condition on the super torus for super abelian variety.

• 정보보호학회논문지
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• 제25권3호
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• pp.567-571
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• 2015

Brezing과 Weng은 페어링 친화 타원곡선의 CM 파라미터들을 수체(number field)의 다항식 표현을 이용하여 생성하는 방법을 제안하였고, Freeman은 그 방법을 아벨 다양체(abelian variety)의 경우로 일반화 시켰다. 본 논문에서는 특히 단순 아벨 곡면(simple abelian surface)의 경우에 대해 Brezing-Weng 방법에서 사용되는 다항식족(polynomial family)을 구하는 새로운 공식들을 유도하고, 이를 이용하여 CM 파라미터들을 생성할 수 있음을 보인다.

• 호남수학학술지
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• 제39권2호
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• pp.137-141
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• 2017

Let A be an abelian variety defined over a number field K and let L be a finite Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Let $Res_{L/K}(A)$ be the restriction of scalars of A from L to K and let B be an abelian subvariety of $Res_{L/K}(A)$ defined over K. Assuming that III(A/L) is finite, we compute [III(B/K)][III(C/K)]/[III(A/L)], where [X] is the order of a finite abelian group X and the abelian variety C is defined by the exact sequence defined over K $0{\longrightarrow}B{\longrightarrow}Res_{L/K}(A){\longrightarrow}C{\longrightarrow}0$.

• 호남수학학술지
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• 제45권3호
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• pp.410-417
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• 2023

Suppose that there are 8 abelian varieties defined over a number field K which satisfy a commutative diagram. We show that if we know that three out of four short exact sequences satisfy the rate formula of Tate-Shafarevich groups, then the unknown short exact sequence satisfies the rate formula of Tate-Shafarevich groups, too.

• 호남수학학술지
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• 제36권2호
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• pp.417-424
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• 2014

Let A be an abelian variety defined over a number field K and p be a prime. Define ${\varphi}_i=(x^{p^i}-1)/(x^{p^{i-1}}-1)$. Let $A_{{\varphi}i}$ be the abelian variety defined over K associated to the polynomial ${\varphi}i$ and let Ш($A_{{\varphi}i}$) denote the Tate-Shafarevich groups of $A_{{\varphi}i}$ over K. In this paper assuming Ш(A/F) is finite, we compute [Ш($A_{{\varphi}1}$)][Ш($A_{{\varphi}2}$)]/[Ш($A_{{\varphi}1{\varphi}2}$)] in terms of K-rational points of $A_{{\varphi}i}$, $A_{{\varphi}1{\varphi}2}$ and their dual varieties, where [X] is the order of a finite abelian group X.

• Korean Journal of Mathematics
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• 제30권3호
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• pp.555-560
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• 2022

Suppose L/K be a finite abelian extension of number fields of odd degree and suppose an abelian variety A defined over L is a K-variety. If the endomorphism algebra of A/L is a field F, the followings are equivalent : (1) The enodomorphiam algebra of the restriction of scalars from L to K is simple. (2) There is no proper subfield of L containing L^G_F on which A has a K-variety descent.

• 호남수학학술지
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• 제38권1호
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• pp.85-93
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• 2016

Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.

• 호남수학학술지
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• 제37권1호
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• pp.1-6
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• 2015

Let A be an abelian variety defined over a number field K. Let L be a biquadratic extension of K with Galois group G and let III (A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming III(A/L) is finite, we compute [III(A/K)]/[III(A/L)] where [X] is the order of a finite abelian group X.

• 대한수학회지
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• 제61권3호
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• pp.515-545
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• 2024

Consider a Noetherian domain R₀ with quotient field K₀. Let K be a finitely generated regular transcendental field extension of K₀. We construct a Noetherian domain R with Quot(R) = K that contains R₀ and embed Spec(R₀) 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 p for "almost all" p ∈ Spec(R₀).