On the fibres of an elliptic surface where the rank does not jump (original) (raw)

On the rank of the fibers of elliptic K3 surfaces

Bulletin of the Brazilian Mathematical Society, New Series, 2012

Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations π i , i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C ⊂ X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of π j is positive then there are infinitely many fibers of π i (j = i) with rank at least the generic rank of π i plus one.

On the extendability of elliptic surfaces of rank two and higher

Annales de l’institut Fourier, 2009

We study threefolds X ⊂ P r having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.

The liftability of elliptic surfaces

2011

v As a conclusion, we can say that, although non-liftable elliptic fibrations do exist, this is not the typical behaviour. Given that p is sufficiently large in comparison to certain arithmetic invariants of the surface in question, liftability is seen to hold true.

Mordell-Weil theorem and the rank of elliptical curves

2007

An elliptic curve over a field is a nonsingular cubic curve in two variables with a rational point over the field. The set of these rational points forms an abelian group by the suitable definition of the group operation. If the field is an algebraic number field, Mordell-Weil theorem states that the group of rational points is finitely generated. The rank of an elliptic curve is the size of the smallest torsion-free generating set. The rank is very important in the study of elliptic curves. The rank is involved with many significant open questions on elliptic curves these days including the Birch and Swinnerton-Dyer Conjecture, which is one of the seven Millennium Prize problems established by the Clay Mathematics Institute. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptic curves in certain cases over algebraic number fields can be obtained and computable. This formula was first observed by J. Tate. The objective of this thesis is to give a detaile...

Sections on certain j=0 elliptic surfaces

Using some theory of (rational) elliptic surfaces plus elementary properties of a Mordell-Weil group regarded as module over the endomorphism ring of a (CM) elliptic curve, we present examples of such surfaces with j-invariant zero. In particular, a K3 example is given with the maximal possible rank 18, plus that many explicit independent sections on it. Several other high rank cases are discussed; for instance Shioda's example with rank 68.

Rational points on certain elliptic surfaces

Acta Arithmetica, 2007

\ Q, and let us assume that deg f ≤ 4. In this paper we prove that if deg f ≤ 3, then there exists a rational base change t → ϕ(t) such that there is a non-torsion section on the surface E f •ϕ . A similar theorem is valid in case when deg f = 4 and there exists t 0 ∈ Q such that infinitely many rational points lie on the curve Et 0 : y 2 = x 3 + f (t 0 )x. In particular, we prove that if deg f = 4 and f is not an even polynomial, then there is a rational point on E f . Next, we consider a surface E g : y 2 = x 3 + g(t), where g ∈ Q[t] is a monic polynomial of degree six. We prove that if the polynomial g is not even, there is a rational base change t → ψ(t) such that on the surface E g•ψ there is a non-torsion section. Furthermore, if there exists t 0 ∈ Q such that on the curve E t 0 : y 2 = x 3 +g(t 0 ) there are infinitely many rational points, then the set of these t 0 is infinite. We also present some results concerning diophantine equation of the form x 2 − y 3 − g(z) = t, where t is a variable.

On the slope of rational fibered surfaces

Osaka Journal of Mathematics, 2020

Given a rational fibered surface f : X → P1 of genus g we prove the inequality 6n+5 n+1 − 9n+12 2g ≤ λ f , provided that the genus g is sufficiently high with respect to the gonality 2n+3 of the general fibre.

Two infinite families of elliptic curves with rank greater than one

2021

We prove, using elementary methods, that each member of the infinite families of elliptic curves given by Em : y = x+x−m and E m : y 2 = x − x+m have rank at least 2 and 3, respectively, under mild assumptions on m. We also prove a stronger result for E m using more technical machinery.