# Schedule of the Workshop

## Thursday, March 11

## Friday, March 12

09:30-10:30 |
Klaus Hulek: Enriques Surfaces and jacobian elliptic K3 surfaces |

11:00-12:00 |
Masato Kuwata: Hessian and the elliptic modular surface for level 3 |

14:00-15:00 |
Valerio Pastro: Construction of linear pencils of cubics with Mordell-Weil rank four |

15:30-16:30 |
Chad Schoen: The geometric Tate Shafarevich group for certain elliptic curves over elliptic surfaces |

17:00-18:00 |
Rene Pannekoek: Parametrizing cubic surfaces and constructing Severi-Brauer surfaces over number fields |

## Abstracts:

Anna Fluder: Asymptotic behavior of elliptic curves over function fields

We study ("soft") upper bounds for the average Mordell-Weil-ranks of elliptic fibrations over curves over finite fields for varying height. We discuss the cases for genus zero (A. J. de Jong) and genus one.

Chad Schoen: The geometric Tate Shafarevich group for certain elliptic curves over elliptic surfaces

In this joint work with Shahed Sharif the geometric Tate-Shafarevich group of certain elliptic curves over the function field of an elliptic surface is computed. I hope to start with a review of the definition and meaning of the Tate-Shafarevich group, to say a few words about the computation, and perhaps about applications.

Klaus Hulek: Enriques Surfaces and jacobian elliptic K3 surfaces

We discuss a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour.

Remke Kloosterman: The average rank of elliptic threefolds

For elliptic curves over number fields it is conjectured that half the curves have rank 1 and half the curves have rank 0. Similarly, if C/IF_{q} is a curve then it is conjectured the half the elliptic curves over IF_{q}(C) have rank 0 and half the curves have rank 1.

In this talk we show that the situation is different if one considers elliptic curves over IF_{q}(V), with dim(V)>1. In particular, we show that the rank 0 elliptic curves over F_{q}(V) have density 1 in case dim V>2. In the two-dimensional case we give some numerical evidence for the same statement.

Rene Pannekoek: Parametrizing cubic surfaces and constructing Severi- Brauer surfaces over number fields

Let S/K be a smooth projective cubic surface in P3 over a perfect field. First I review a known necessary and sufficient criterion for the existence of a K-birational map from P2 to S. Also, assuming that no birational map exists, I show that a K-rational map with degree at most 6 still exists under the condition that S(K) is non-empty and K has cardinality at least 37. This has all been known since the 70s and follows from work by Manin and Swinnerton-Dyer. Let now S/K be an explicitly given cubic curface as above, and K a number field. I will then show how to verify in practice the existence of a birational map from P2 to S. Lastly, I will construct two cubic surfaces over Q which satisfy the said criterion but for the existence of a Q-rational point. This comes down to constructing an explicit Severi-Brauer surface over Q, for which I will also outline a method.

Holger Partsch: Deformations of elliptic fibre bundles in positive characteristic

We study deformations over the Witt vectors of elliptic fibrations of the simplest kind:

The locally trivial ones.

Bi-elliptic surfaces are our chief examples. Special attention will be paid to those cases, where the order of monodromy is divided by p, because this leads to obstructions against lifting to characteristic zero.

Valerio Pastro: Construction of linear pencils of cubics with Mordell-Weil rank four

The purpose is to construct Rational Elliptic Surfaces with Mordell-Weil rank four, starting from a linear pencil of cubics in IP^{2}.

The construction is similar to Salgado's and Fusi's for higher ranks, but we have to pay attention to a new feature: in the previously studied cases the Mordell-Weil group had no torsion; in our setting this can occur.

A rational elliptic surface can be obtained as the blow-up of IP^{2} at nine points, precisely the base points of a pencil of cubics in IP^{2}. The configuration of those points determines the type of the reducible fibers in the surface and more generically the rank.

We use Shioda and Oguiso's results on the Mordell-Weil lattices on a rational elliptic surface; we restric to the rank 4 case and treat each of the possible cases for the Mordell-Weil lattice separately (giving the constructions in each case as well as computing the heights of its generators).

Stefan Schröer: Enriques manifolds

Using the theory of hyperkahler manifolds, we generalize the notion of Enriques surfaces to higher dimensions and construct severalexamples using groups actions on Hilbert schemes of points or modulispaces of stable sheaves.

Ronald van Luijk: Density of rational points on elliptic surfaces

Let X be a diagonal quartic surface in projective threespace over the rational numbers of which the product of the coefficients is a square. It is true that if X contains a rational point outside the 48 lines on X and outside the four coordinate planes, then the set of rational points on X is Zariski dense. The main ingredient in the proof is the fact that X admits two different elliptic fibrations. In this talk we will generalize a generalization of this statement by Swinnerton-Dyer.