Related papers: On cubic elliptic varieties

\special{html:<a href="hrefstring">} Let $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that…

Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$ The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset V(k)$ such that the whole $V(k)$ can be obtained from $P$ by drawing secants and…

For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the…

We describe in terms of the j-invariant all elliptic surfaces pi: X -> C with a section, such that h^{1,1}(X)=rank NS(X) and the Mordell-Weil group of pi is finite. We use this to give a complete solution to infinitesimal Torelli for…

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…

This is mainly a review of my results related to the title. We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically closed…

We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such…

This is a paper in smooth $4$-manifold topology, inspired by the N\'{e}ron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(\pi)$ of Mordell-Weil group of an elliptic fibration $\pi:M\to\Pb^1$ is finitely…

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of $\Q$, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite…

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

We classify all the possible configurations of singular fibers and the torsion parts of Mordell-Weil groups of complex elliptic K3 surfaces. The complete list of 3279 configurations is attached.

The geometry of elliptic fibrations translates to the physics of gauge theories in F-theory. We systematically develop the dictionary between arithmetic structures on elliptic curves as well as desingularized elliptic fibrations and…

Let $A$ be an abelian variety over a $p$-adic field $K$ and $L$ an algebraic infinite extension over $K$. We consider the finiteness of the torsion part of the group of rational points $A(L)$ under some assumptions. In 1975, Hideo Imai…

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of…

We present two proofs for a bound on the rank of the Mordell-Weil group of some elliptic fibrations. The bounds apply to Calabi-Yau varieties, which are also of interest to the physics of string theory. We prove explicit bounds for…

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the…

Let $X$ be a smooth projective surface over a number field $K$. Assume that $X$ has an elliptic fibration over $\mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/U$ be a smooth projective model of $X$ over…

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which…