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Remke Kloosterman

Publications and source records attributed to Remke Kloosterman.

At least 37 records · Page 2Linked to original sources
arXiv2013

On the density of abelian surfaces with Tate- Shafarevich group of order five times a square

Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has square order or order five times a square, assuming that we can find a basis for the Mordell-Weil groups of both E_i, and that the Tate-Shafarevich groups of the E_i are finite. We considered all pairs (E_1,E_2), with prescribed bounds on the conductor and the coefficients on a minimal Weierstrass equation. In total we considered around 20.0 million of abelian surfaces of which 49.16% have a Tate-Shafarevich group of non-square order.

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arXiv2012

Cuspidal plane curves, syzygies and a bound on the MW-rank

Let $C=Z(f)$ be a reduced plane curve of degree $6k$, with only nodes and ordinary cusps as singularities. Let $I$ be the ideal of the points where $C$ has a cusp. Let $\oplus S(-b_i)\to \oplus S(-a_i) \to S\to S/I$ be a minimal resolution of $I$. We show that $b_i\leq 5k$. From this we obtain that the Mordell-Weil rank of the elliptic threefold $W:y^2=x^3+f$ equals $2#\{i\mid b_i=5k\}$. Using this we find an upper bound for the Mordell-Weil rank of $W$, which is $1/18 (125+\sqrt{73}-\sqrt{2302-106\sqrt{73}})k+l.o.t.$ and we find an upper bound for the exponent of $(t^2-t+1)$ in the Alexander polynomial of $C$, which is $1/36(125+\sqrt{73}-\sqrt{2302-106\sqrt{73}})k+l.o.t.$. This improves a recent bound of Cogolludo and Libgober almost by a factor 2.

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arXiv2012

A structure theorem for fibrations on Delsarte surfaces

In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change the fibration is the pull back of a fibration with three singular fibers, and that this second base change is completely ramified at two points where the fiber is singular. As a corollary we show that every Delsarte fibration of genus 1 with nonconstant $j$-invariant occurs as the base change of an elliptic surface from Fastenberg's list of rational elliptic surfaces with $γ<1$.

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arXiv2011

Mordell-Weil groups and Zariski triples

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.

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arXiv2011

The average rank of elliptic $n$-folds

Let $V/\mathbb{F}_q$ be a variety of dimension at least two. We show that the density of elliptic curves $E/\mathbb{F}_q(V)$ with positive rank is zero if $V$ has dimension at least 3 and is at most $1-ζ_V(3)^{-1}$ if $V$ is a surface.

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arXiv2010

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

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arXiv2009

Zeta-functions of certain K3 fibered Calabi--Yau threefolds

We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.

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arXiv2009

On the classification of degree 1 elliptic threefolds with constant $j$-invariant

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in $\Ps(2,3,1,1,1)$ that admit no variation of Hodge structure.

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arXiv2008

The zeta-function of monomial deformations of Fermat hypersurfaces

This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in P^n. In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the p-adic Picard-Fuchs equation associated with monomial deformations of Fermat hypersurfaces. As a by-product we obtain Poincare Duality for the Rigid cohomology of certain singular affine varieties.

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arXiv2006

The L-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface $S$, defined over a number field K such that S^[2](K) is not empty, then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.

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arXiv2006

Modularity of Calabi-Yau varieties

In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.

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arXiv2005

Elliptic K3 surfaces with geometric Mordell-Weil rank 15

We prove that the elliptic surface y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1) has geometric Mordell-Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic K3-surfaces with geometric Mordell-Weil rank 0,1,..., 14, 16, 17,18.

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