# The Arithmetic of Fields

### Wulf-Dieter Geyer

Universität Erlangen-Nünberg, Germany### Moshe Jarden

Tel Aviv University, Israel### Florian Pop

University of Pennsylvania, Philadelphia, United States

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## Abstract

The fifth conference with the title {\bf The Arithmetic of Fields}, organized by Wulf-Dieter Geyer (Erlangen), Moshe Jarden (Tel Aviv), and Florian Pop (Philadelphia), was held February 5--11th, 2006. In contrast to the fourth conference held in February 3--9th, 2002, this conference was a 'full' one, namely as many participants were invited as the Institute could host. Due to support from the European Union, more young people were invited in the last few weeks prior to the conference, so that the total number of participants reached 54. The participants came from 13~~countries: Germany~~(20), USA~(10), Israel~(7), France~(7), Denmark~(2), Austria~(1), Brazil~(1), Canada~(1), Hungary~(1), Japan~(1), Romania~(1), Russia~(1), and South Africa~(1). Among the participants there were 9 graduate students and 8 young researchers. Six women attended the conference.

The organisers asked four people before the conference to give surveys of one hour on recent progress made by other colleagues in Field Arithmetic.

Tam\'as Szamuely (Budapest) reported on the solution by J\'anos Koll\'ar of a Problem due to Ax from 1968: Every PAC field of characteristic $0$ is $C_1$. The case where the characteristic is positive remains open.

Alexandra Shlapentokh (Greenville) described the progress Bjorn Poonen made on Hilbert's Tenth Problem: There exists a recursive set $T_1$ of prime numbers of natural density $0$ and a set $T_2$ of prime numbers of natural density $1$ such that $T_1\subseteq T_2$ and for each set $S$ with $T_1\subseteq S\subseteq T_2$ Hilbert's Tenth Problem for $O_{\bbQ,S}$ has a negative solution. Here $O_{\bbQ,S}$ is the ring of all rational numbers whose denominators are divided only by primes in $S$. Whether Hilbert's Tenth Problem for $\bbQ$ has a negative solution is still open.

Alexander Prestel (Konstanz) presented a theorem of Jochen Koenigsmann: If a $p$-Sylow extension $P$ of a field $K$ is Henselian and $P$ is neither separably closed nor real closed, then $K$ itself is Henselian.

Pierre D\`ebes (Lille) surveyed Fried's problem on Modular Towers. He mentioned that the main conjecture is close to completion in the case of $4$ branch points (Bayley-Fried). He also reported on a result of Anna Cadoret: The dihedral group has a regular realization over $\bbQ_{\rm ab}$ with only inertia groups of order~$2$.

In addition to these survey talks seventeen participants were invited to report on their own achievements in 45 minutes talks. Altogether, the talks presented the impressive progress made in Field Arithmetic in recent years. The reader may find here extended abstracts of all talks. We hope they will be to the benefit of all of the participants as well as the fans of Field Arithmetic.

The organisers: Wulf-Dieter Geyer, Moshe Jarden, Florian Pop