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ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS
( Hoseog Yu ) 호남수학회 2016 호남수학학술지 Vol.38 No.1
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 Ш (A/K) and Ш (A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that Ш (A/L) is finite, we compute [Ш (A/K)][ Ш (Aφ/K)]/[ Ш (A/L)], where [X] is the order of a finite abelian group X.
ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS
Yu, Hoseog The Honam Mathematical Society 2015 호남수학학술지 Vol.37 No.1
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.
ON THE RATIO OF TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS OF ORDER p<sup>2</sup>
Yu, Hoseog The Honam Mathematical Society 2014 호남수학학술지 Vol.36 No.2
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.
ON THE RATIO OF TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS OF ORDER P(2)
( Hoseog Yu ) 호남수학회 2014 호남수학학술지 Vol.36 No.2
Let A be an abelian variety defined over a number field K and p be a prime. Define φi = (x(p(i) - 1))/(x(p(i-1) - 1)). Let Aφi be the abelian variety defined over K associated to the poly-nomial φi and let Ш(Aφi) denote the Tate-Shafarevich groups of Aφi over K. In this paper assuming Ш(A/F) is finite, we compute [Ш(Aφ1)][Ш(Aφ2)]/[\Ш(Aφ1φ2)] in terms of K-rational points of Aφi, Aφ1φ2 and their dual varieties, where [X] is the order of a finite abelian group X.
ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS
Yu, Hoseog The Honam Mathematical Society 2016 호남수학학술지 Vol.38 No.1
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.
TATE-SHAFAREVICH GROUPS AND SCHANUEL'S LEMMA
Yu, Hoseog The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.2
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 $0{\longrightarrow}B{\longrightarrow}Res_{L/K}(A){\longrightarrow}C{\longrightarrow}0$.
TATE-SHAFAREVICH GROUPS AND SCHANUEL`S LEMMA
( Hoseog Yu ) 호남수학회 2017 호남수학학술지 Vol.39 No.2
Let A be an abelian variety defined over a number field K and let L be a finite Galois extension of K. Let Щ(A/K) and Щ(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Let Res<sub>L/K</sub>(A) be the restriction of scalars of A from L to K and let B be an abelian subvariety of Res<sub>L/K</sub>(A) defined over K. Assuming that Щ(A/L) is finite, we compute [Щ(B/K)][Щ(C/K)]/[Щ(A/L)], where [X] is the order of a finite abelian group X and the abelian variety C is de_ned by the exact sequence de_ned over K 0 → B → Res<sub>L/K</sub>(A) → C → 0.
TATE-SHAFAREVICH GROUPS OVER THE COMMUTATIVE DIAGRAM OF 8 ABELIAN VARIETIES
( Hoseog Yu ) 호남수학회 2023 호남수학학술지 Vol.45 No.3
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.
Completely <i>p</i>-primitive binary quadratic forms
Oh, Byeong-Kweon,Yu, Hoseog Elsevier 2018 Journal of number theory Vol.193 No.-
<P><B>Abstract</B></P> <P>Let f ( x , y ) = a <SUP> x 2 </SUP> + b x y + c <SUP> y 2 </SUP> be a binary quadratic form with integer coefficients. For a prime <I>p</I> not dividing the discriminant of <I>f</I>, we say <I>f</I> is completely <I>p</I>-primitive if for any non-zero integer <I>N</I>, the diophantine equation f ( x , y ) = N always has an integer solution ( x , y ) = ( m , n ) with ( m , n , p ) = 1 whenever it has an integer solution. In this article, we study various properties of completely <I>p</I>-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form <I>f</I> to be completely <I>p</I>-primitive.</P>
New infinite families of 3-designs from algebraic curves over <sub>Fq</sub>
Oh, Byeong-Kweon,Oh, Jangheon,Yu, Hoseog Elsevier 2007 European journal of combinatorics : Journal europ& Vol.28 No.4
<P><B>Abstract</B></P><P>In this paper, we show that the stabilizer subgroup of Df+={a∈<SUB>Fq</SUB>|f(a)∈<SUP>(Fq×)2</SUP>} for a f∈<SUB>Fq</SUB>[x] without multiple roots can be derived from the stabilizer of Df0={a∈<SUB>F¯q</SUB>|f(a)=0}∪{∞}. As an application, we construct a family of 3-designs such as 3−(q+1,q−12,(q−1)(q−3)(q−5)16), where q is a prime power such that q≡3(mod4) and q≥59.</P>