Andreas-Stephan Elsenhans

On the computation of the Picard group for K3 surfaces

We present a method to construct examples of K 3 surfaces of geometric Picard rank 1. Our approach is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on etale cohomology. This allows us to abandon the original limitation to cases of Picard rank 2 after reduction modulo p . Furthermore, the use of Galois data enables us to construct examples that require significantly less computation time.

K3 surfaces of Picard rank one and degree two

We construct explicit examples of K3 surfaces over Q whichare of degree 2 and geometric Picard rank 1. We construct, particularly,examples of the form w2 = det M where M is a (3 × 3)-matrix of ternaryquadratic forms.

Experiments with General Cubic Surfaces

For general cubic surfaces, we test numerically the conjecture of Manin (in the refined form due to E. Peyre) about the asymptotics of points of bounded height on Fano varieties. We also study the behavior of the height of the smallest rational point versus the Tamagawa type number introduced by Peyre.

Cubic surfaces with a Galois invariant double-six

We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.

Kummer surfaces and the computation of the Picard group

We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in P 3 . Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when suciently

Cubic surfaces with a Galois invariant pair of Steiner trihedra

We present a method to construct non-singular cubic surfaces over

On Weil Polynomials of K3 Surfaces

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Invariants for the computation of intransitive and transitive Galois groups

with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of A. Cayley and G. Salmon. For these, we develop an explicit version of Galois descent.

The asymptotics of points of bounded height on diagonal cubic and quartic threefolds

For K3 surfaces, we derive some conditions the characteristic polynomial of the Frobenius on the etale cohomology must satisfy. These conditions may be used to speed up the computation of Picard numbers and the decision of the sign in the functional equation**. Our investigations are based on the Artin-Tate formula.

Moduli spaces and the inverse Galois problem for cubic surfaces

Abstract One hard step in the computation of Galois groups by Stauduhar’s method is the construction of relative invariants. In this note, a representation-theoretic approach is given for the construction in the case of an intransitive group. In the second part of the article, it is shown that the construction can be used for groups that have a suitable intransitive subgroup. The construction solves an open question of Fieker and Kluners.