Remke Kloosterman

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g + 1 over the rationals, where g is the genus of X-0(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p = 5, 7 and 13 (the cases p less than or equal to 7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N = 9, 10, 12, 16 and 25

Elliptic K3 Surfaces with Geometric Mordell-Weil Rank 15

We prove that the elliptic surface y2 = x3 + 2(t8 + 14t4 + 1) x + 4t2 (t8 + 6t4 + 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 ranks 0, 1, dots, 14, 16, 17, 18.

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.

ZETA-FUNCTIONS OF CERTAIN K3-FIBERED CALABI–YAU THREEFOLDS

We consider certain K3-fibered Calabi–Yau threefolds. One class of such Calabi–Yau threefolds is 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 K3-fibered Calabi–Yau threefolds. In particular, we give detailed discussions on the construction of these Calabi–Yau varieties, singularities and their resolutions. We then determine the zeta-functions of these Calabi–Yau varieties. Next we consider deformations of our K3-fibered Calabi–Yau threefolds, and we study the variation of the zeta-functions using p-adic rigid cohomology theory.

Locally trivial families of hyperelliptic curves: The geometry of the Weierstrass scheme

Abstract In this paper we describe some geometrical properties of the Weierstrass scheme of locally trivialhyperelliptic fibrations.

Point counting on singular hypersurfaces

Let q = p r be a prime power. Let Open image in new window be a homogenous polynomial of degree d. Let Open image in new window be the hypersurface defined by Open image in new window . A natural question to ask is how to determine Open image in new window .

Algebraic and Complex Geometry: In Honour of Klaus Hulek’s 60th Birthday

Preface.- Introduction.- A. Fruhbis-Kruger, R. Kloosterman, M. Schutt: Introduction.- M.A. Barja, L. Stoppino: Stability conditions and positivity of invariants of fibrations.- A. Beauville: On the second lower quotient of the fundamental group.- M. Blume: McKay correspondence over non algebraically closed fields.- L. Caporaso: Gonality of algebraic curves and graphs.- F. Catanese: Caustics of plane curves, their birationality and matrix projections.- O. Fujino, Y. Gongyo: On images of weak Fano manifolds II.- N. Hitchin: The hyperholomorphic line bundle.- H. Hollborn, S. Muller-Stach: Hodge numbers for the cohomology of Calabi-Yau type local systems.- E. Markman: Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type.- T. Peternell, F. Schrack: Contact Kahler Manifolds: Symmetries and Deformations.- Complete List of talks.- List of participants.

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 pullback 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.