Bo‐Hae Im

Abstract Let K be a number field, an algebraic closure of K and E / K an elliptic curve defined over K . In this paper, we prove that if E / K has a K -rational point P such that 2 P ≠ O and 3 P ≠ O , then for each σ ∈ Gal( / K ), the Mordell–Weil group of E over the fixed subfield of under σ has infinite rank.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E slash double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E/\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an e

Let $K$ be a field of characteristic $\neq 2$ such that every finite separable extension of $K$ is cyclic. Let $A$ be an abelian variety over $K$. If $K$ is infinite, then $A(K)$ is Zariski-dense in $A$. If $K$ is not locally finite, the rank of $A$ over $K$ is infinite.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K overbar"> <mml:semantics> <mml:mover> <mml:mi>K</mml:mi> <mml:mo accent="false"> ¯ </mml:mo>

Bo-Hae Im

In this paper, we give examples of elliptic curves E/K over a number field K satisfying the property that there exist P1, P2 ∈ K[t] such that the twists E P 1 , E P 2 and E P 1 P 2 are of positive rank over K(t). As a consequence of this result on twists, we show that for those elliptic curves E/K, and for each σ ∈ G a l ( K ¯ / K ) , the rank of E over the fixed field (Kab)σ under σ is infinite, where Kab is the maximal abelian extension of K.

If the system of two diophantine equations $X^2+mY^2=Z^2$ and $X^2+nY^2=W^2$ has infinitely many integer solutions $(X,Y,Z,W)$ with $\operatorname {gcd}(X,Y)=1$, equivalently, the elliptic curve $E_{m,n} : y^2=x(x+m)(x+n)$ has positive rank over $\mathbb {Q}$, then $(m,n)$ is called a strongly concordant pair. We prove that for a given positive integer $M$ and an integer $k$, the number of strongly concordant pairs $(m, n)$ with $m,n\in [1,N]$ and $m,n \equiv k$ is at least $O(N)$, and we give a

Abstract For a square-free positive integer N , we study the normalizer of Γ Δ ( N ) in PSL 2 (ℝ) and investigate the group structure of its quotient by Γ Δ ( N ) under certain conditions.

If $E$ is an elliptic curve defined over a quadratic field $K$, and the $j$-invariant of $E$ is not $0$ or $1728$, then $E(\mathbb{Q}^{\mathrm{ab}})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-\lambda)$, where $\math

We give conditions for when the fixed points by the partial Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points as an extension of the result by Lehner and Newman [18]. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points or not.