相关论文: Structure of Tate-Shafarevich groups of elliptic c…

Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$…

Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi -$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x…

This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic…

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate…

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…

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we find Zariski…

Let $k_0$ be a number field, $K$ be a finite extension of $k_0(\!(x_1,...,x_n)\!)$ and let $R$ be the integral closure of $k_0[[x_1,...,x_n]]$ in $K$. Consider a group of multiplicative type $G$ defined over $K$. We study the…

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to…

We study Witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two. In both dimensions, we determine both the classical Witt group and Balmer's shifted Witt groups. In the case of curves, the…

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…

The Directed Power Graph of a group is a graph whose vertex set is the elements of the group, with an edge from $x$ to $y$ if $y$ is a power of $x$. The \textit{Power Graph} of a group can be obtained from the directed power graph by…

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the…

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all…

For $s >\frac{3}{2}$, the group of Sobolev class s diffeomorphisms of the circle is a smooth manifold modeled on the space of Sobolev class s sections of the tangent bundle of the circle. It is a topological group in the sense that…

Let E be an elliptic curve over Q with complex multiplication. The aim of the present paper is to strengthen the theoretical and numerical results of \cite{CZS}. For each prime p, let t_{E/Q, p} denote the Z_p-corank of the p-primary…

Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$…

Let $E/\mathbb{Q}$ be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of $E/\mathbb{Q}$ divides the…

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group, whose graded traces are weight 3/2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to…