We classify elliptic curves over the rationals whose Néron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we describe those curves where the element of order two is narrow, or where another element of order two exists, and also express our findings in terms of Deligne--Mumford stacks of pointed curves of genus one.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves…
We derive an efficient algorithm to find solutions to Euler's concordant form problem and rational points on elliptic curves associated with this problem.
We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison-Kawamata cone conjecture holds for these nef cones.
We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.
Let $X$ be a rational elliptic surface with elliptic fibration $π:X\to\Bbb{P}^1$ over an algebraically closed field $k$ of any characteristic. Given a conic bundle $φ:X\to\Bbb{P}^1$ we use numerical arguments to classify all possible fibers of $φ$ and study the interplay between singular fibers of $π$ and $φ$.
We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any elliptic K3 surface. When the characteristic of $k$ is zero, this result is due to the work of Bogomolov-Tschinkel and Hassett.
We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}(Ω_X)$ is dense in the Zariski topology. As an application we give a simple proof of a theorem of Kobayashi in the elliptic case, i.e., there are no globally defined symmetric differential forms.
This paper deals with a study of the rational elliptic surfaces whose $J$-invariant functions are of degree one. Almost all of these elliptic surfaces have four singular fibers, while the remaining surfaces have only three singular fibers. The moduli space of these elliptic surfaces is canonically isomorphic to the projective line by taking the $J$-values for a certain fixed type of singular fibers. Over the moduli space, we discuss our elliptic surfaces, and investigate how their sections are described by the parameter of the moduli space. By using a covering space of the moduli, we construct…
In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to study the geometry of sections. As a result, we are able to give new proofs for some classical results on singular plane quartics and their bitangent lines.
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron-Severi lattices. Over a field of characteristic $0$, we prove that the existe…
Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves E/Q such that Q(E[n]) is as small as possible, that is, when Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5. More generally, we classify all curves such that Q(E[n]) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem), when Q(E[n])/Q is an abelian extension. In particular, we prove that…
Given two elliptic curves, each of which is associated with a projection map that identifies opposite elements with respect to the natural group structure, we investigate how their corresponding projective images of torsion points intersect.
Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Department of Mathematics, Faculty of Science, Kyoto University（京都大学理学部数学科）；Department of Mathematics, Cooperative Faculty of Education, Gunma University（群马大学教育学部协同数学科）；Department of Mathematics, Graduate School of Science, Kobe University（神户大学理学研究数理学科）；Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University（东京都立大学理学研究科数学科学系）
This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.
We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over $\mathbb{Q}$.
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which sp…
Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense set of $\mathbb{Q}$-rational points. In this paper we work on solving the conjecture in case $\mathscr{E}$ is rational by means of geometric and analytic methods. First, we show that for $\mathscr{E}$ rational, the set $\mathscr{E}(\mathbb{Q})$ is Zariski-dense when $\mathscr{E}$ is isotrivial with non-zero $j$-invariant and when $\mathscr{E}$ is non-isotrivi…
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem regarding the set of fibers whose Mordell-Weil ranks are at least 3 units higher than the generic rank of the elliptic fibration.
We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields $\mathbb{F}_q$ with a prescribed (small) group $E(\mathbb{F}_q)$. In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .
Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified covers of abelian varieties, and Faltings's finiteness theorem for rational points on higher genus curves.
In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
