Related papers: On signed Mordell-Weil groups for abelian varietie…

We study the boundedness of the Mordell-Weil rank and the growth of the $v$-primary part of the Tate-Shafarevich group of $p$-supersingular abelian varieties of ${\rm GL}_2$-type with real multiplication over $\mathbb{Z}_p$-extensions of…

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer…

In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption…

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$.…

We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\'eron-Tate height of generators of its Mordell-Weil group. The…

We study the \Lambda-module structure of the Mordell-Weil, Selmer, and Tate-Shafarevich groups of an abelian variety over \mathbb{Z}_p-extensions.

Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a…

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and…

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we…

Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's…

We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To…

We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove…

Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a…

For a given number field $K$, we show that the ranks of nonsingular elliptic curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce Nonstandard…

In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic extensions of a wide range of abelian varieties defined over a number field $F$ whose primes above $p$ are totally ramified over $F/\mathbb{Q}$. We assume that the…

Given an elliptic curve defined over a number field $F$, we study the algebraic structure and prove a control theorem for Wuthrich's fine Mordell--Weil groups over a $\mathbb{Z}_p$-extension of $F$, generalizing results of Lee on the usual…

Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has…