Arithmetic of the [19, 1, 1, 1, 1, 1] fibration (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 uniqueness of elliptic K3 surfaces with maximal singular fibre

Annales de l’institut Fourier, 2013

We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, I 19 and I * 14 , the surface is unique. In characteristic 2 the maximal fibre types are I 18 and I * 13 , and there exist two (resp. one) one-parameter families of such surfaces.

Miranda-Persson’s problem on extremal elliptic K3 surfaces

2002

Let f : X → C be an elliptic surface over a smooth projective curve C with a section O, i.e., a Jacobian elliptic fibration over C. Throughout this paper, we always assume that (*) f has at least one singular fiber. Let M W (f) be the Mordell-Weil group of f : X → C, i.e., the group of sections, O being the zero. Under the assumption (*), it is known that M W (f) is a finitely generated abelian group (the Mordell-Weil theorem). More precisely, if we let R be the subgroup of the Néron-Severi group, NS(X), of X generated by O and all the irreducible components in fibers of f , then (i) NS(X) is torsion-free, and (ii) M W (f) ∼ = NS(X)/R (see [S], for instance). Note that the Shioda-Tate formula rank M W (f) = ρ(X) − rank R easily follows from the second statement. We call f : X → C extremal if (i) the Picard number ρ(X) of X is equal to h 1,1 and (ii) rank M W (f) = 0. If f : X → C is extremal, then the Shioda-Tate formula implies rank R = ρ(X). Hence, in other words, f : X → C is extremal if and only if ρ(X) = rank R = h 1,1 (X). Also, taking the isomorphism M W (f) ∼ = N S(X)/R into account, it seems that we can say much about M W (f) only from the data of types of singular fibers. For extremal rational elliptic surfaces, Miranda and Persson studied them from several viewpoints [MP1]; and for such surfaces, M W (f) is determined by the data of types of singular fibers. Moreover, they proved 1 Partially supported by CAICYT PB94-0291 and DGES PB97-0284-C02-02 2 Research partly supported by the Grant-in-Aid for Encouragement of Young Scientists 09740031 from the Ministry of Education, Science and Culture 3 Financial support by the JSPS-NUS program

K3 Surfaces Associated with Curves of Genus Two

International Mathematics Research Notices, 2010

It is known ([10], [27]) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over P 1 with specified singular fibers of type II * and III *. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C.

24 rational curves on K3 surfaces

2019

Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p different from 2,3) of sufficiently high degree contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. We also show that the bounds are sharp and attained only by K3 surfaces with genus one fibrations.

Simultaneous twists of elliptic curves and the Hasse principle for certain K3 surfaces

2016

Simultaneous twists of elliptic curves and the Hasse principle for certain K3 surfaces Vivek Pal In this thesis we unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves, only with places of good or additive reduction. More generally we prove that, given finitely many such elliptic curves defined over a number field (with rational 2-torsion and satisfying some mild conditions) there exists an explicit quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. Our approach to the Hasse Principle is outlined below and was introduced by Skorobogatov and Swinnerton-Dyer in [Skorobogatov and Swinnerton-Dyer, 2005]. We also generalize the result proved in [Skorobogatov and Swinnerton-Dyer, 2005]. If each elliptic curve has a distinct multiplicative place of bad reduction, then we find a quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.

On the fibres of an elliptic surface where the rank does not jump

arXiv (Cornell University), 2022

For a non-constant elliptic surface over P 1 defined over 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 elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.